# lfunc_search downloaded from the LMFDB on 27 June 2026. # Search link: https://www.lmfdb.org/L/1/145/145.143 # Query "{'degree': 1, 'conductor': 145}" returned 81 lfunc_searchs, sorted by root analytic conductor. # Each entry in the following data list has the form: # [Label, $\alpha$, $A$, $d$, $N$, $\chi$, $\mu$, $\nu$, $w$, prim, arith, $\mathbb{Q}$, self-dual, $\operatorname{Arg}(\epsilon)$, $r$, First zero, Origin] # For more details, see the definitions at the bottom of the file. "1-145-145.102-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.102" [[0, 0.0]] [] 0 true true false false 0.17454220557092115 0 1.31275963711 ["Character/Dirichlet/145/102"] "1-145-145.108-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.108" [[0, 0.0]] [] 0 true true false false 0.38444853247059835 0 1.80798748355 ["Character/Dirichlet/145/108"] "1-145-145.109-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.109" [[0, 0.0]] [] 0 true true false false 0.2711252997588437 0 1.78035526199 ["Character/Dirichlet/145/109"] "1-145-145.113-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.113" [[0, 0.0]] [] 0 true true false false -0.1741258417750659 0 0.868080997396 ["Character/Dirichlet/145/113"] "1-145-145.118-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.118" [[0, 0.0]] [] 0 true true false false -0.17454220557092115 0 1.10211906621 ["Character/Dirichlet/145/118"] "1-145-145.12-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.12" [[0, 0.0]] [] 0 true true false false 0.11838383098510252 0 1.91500437835 ["Character/Dirichlet/145/12"] "1-145-145.127-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.127" [[0, 0.0]] [] 0 true true false false 0.36880019478095777 0 2.62069474238 ["Character/Dirichlet/145/127"] "1-145-145.128-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.128" [[0, 0.0]] [] 0 true true false false -0.05782436018968083 0 1.40107555267 ["Character/Dirichlet/145/128"] "1-145-145.129-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.129" [[0, 0.0]] [] 0 true true false false 0.17726754290898195 0 1.8847834665 ["Character/Dirichlet/145/129"] "1-145-145.133-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.133" [[0, 0.0]] [] 0 true true false false -0.11838383098510252 0 1.16799328945 ["Character/Dirichlet/145/133"] "1-145-145.137-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.137" [[0, 0.0]] [] 0 true true false false -0.4393432763546183 0 0.982877324889 ["Character/Dirichlet/145/137"] "1-145-145.139-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.139" [[0, 0.0]] [] 0 true true false false 0.3075258750519966 0 2.25161441112 ["Character/Dirichlet/145/139"] "1-145-145.142-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.142" [[0, 0.0]] [] 0 true true false false 0.002082349399717449 0 2.08001806281 ["Character/Dirichlet/145/142"] "1-145-145.143-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.143" [[0, 0.0]] [] 0 true true false false 0.15578014320872263 0 1.6512145381 ["Character/Dirichlet/145/143"] "1-145-145.144-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.144" [[0, 0.0]] [] 0 true true true true 0.0 0 2.20800367343 ["Character/Dirichlet/145/144"] "1-145-145.17-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.17" [[0, 0.0]] [] 0 true true false false 0.05782436018968083 0 1.76058535081 ["Character/Dirichlet/145/17"] "1-145-145.18-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.18" [[0, 0.0]] [] 0 true true false false 0.4393432763546183 0 3.01894055002 ["Character/Dirichlet/145/18"] "1-145-145.2-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.2" [[0, 0.0]] [] 0 true true false false 0.0016659856038622393 0 1.17325064771 ["Character/Dirichlet/145/2"] "1-145-145.24-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.24" [[0, 0.0]] [] 0 true true false false -0.3075258750519966 0 1.42096364969 ["Character/Dirichlet/145/24"] "1-145-145.27-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.27" [[0, 0.0]] [] 0 true true false false 0.331988334383506 0 0.00559357895984 ["Character/Dirichlet/145/27"] "1-145-145.3-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.3" [[0, 0.0]] [] 0 true true false false -0.036492749451556554 0 1.99172721783 ["Character/Dirichlet/145/3"] "1-145-145.32-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.32" [[0, 0.0]] [] 0 true true false false 0.1397154417232268 0 1.29147269885 ["Character/Dirichlet/145/32"] "1-145-145.34-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.34" [[0, 0.0]] [] 0 true true false false 0.09385775684986175 0 0.905591115789 ["Character/Dirichlet/145/34"] "1-145-145.37-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.37" [[0, 0.0]] [] 0 true true false false -0.19259200360617437 0 0.199696223416 ["Character/Dirichlet/145/37"] "1-145-145.4-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.4" [[0, 0.0]] [] 0 true true false false -0.2711252997588437 0 0.957793295468 ["Character/Dirichlet/145/4"] "1-145-145.43-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.43" [[0, 0.0]] [] 0 true true false false -0.331988334383506 0 2.67230759498 ["Character/Dirichlet/145/43"] "1-145-145.47-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.47" [[0, 0.0]] [] 0 true true false false -0.38444853247059835 0 0.899879037981 ["Character/Dirichlet/145/47"] "1-145-145.48-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.48" [[0, 0.0]] [] 0 true true false false 0.002082349399717449 0 1.82753096526 ["Character/Dirichlet/145/48"] "1-145-145.49-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.49" [[0, 0.0]] [] 0 true true false false -0.22949229632473578 0 0.73475961967 ["Character/Dirichlet/145/49"] "1-145-145.54-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.54" [[0, 0.0]] [] 0 true true false false 0.3577604532264366 0 2.63352361891 ["Character/Dirichlet/145/54"] "1-145-145.64-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.64" [[0, 0.0]] [] 0 true true false false -0.09385775684986175 0 1.34814451646 ["Character/Dirichlet/145/64"] "1-145-145.68-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.68" [[0, 0.0]] [] 0 true true false false -0.1397154417232268 0 0.697221220963 ["Character/Dirichlet/145/68"] "1-145-145.72-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.72" [[0, 0.0]] [] 0 true true false false -0.15578014320872263 0 1.42852421243 ["Character/Dirichlet/145/72"] "1-145-145.73-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.73" [[0, 0.0]] [] 0 true true false false 0.0016659856038622393 0 1.87596768728 ["Character/Dirichlet/145/73"] "1-145-145.74-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.74" [[0, 0.0]] [] 0 true true false false 0.22949229632473578 0 2.07934506788 ["Character/Dirichlet/145/74"] "1-145-145.77-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.77" [[0, 0.0]] [] 0 true true false false 0.1741258417750659 0 1.52806197015 ["Character/Dirichlet/145/77"] "1-145-145.8-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.8" [[0, 0.0]] [] 0 true true false false -0.36880019478095777 0 1.29499788792 ["Character/Dirichlet/145/8"] "1-145-145.9-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.9" [[0, 0.0]] [] 0 true true false false -0.17726754290898195 0 1.21455985776 ["Character/Dirichlet/145/9"] "1-145-145.94-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.94" [[0, 0.0]] [] 0 true true false false -0.3577604532264366 0 0.988635556486 ["Character/Dirichlet/145/94"] "1-145-145.97-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.97" [[0, 0.0]] [] 0 true true false false 0.036492749451556554 0 2.11668081154 ["Character/Dirichlet/145/97"] "1-145-145.98-r0-0-0" 0.673377169063117 0.673377169063117 1 145 "145.98" [[0, 0.0]] [] 0 true true false false 0.19259200360617437 0 1.84231011182 ["Character/Dirichlet/145/98"] "1-145-145.103-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.103" [[1, 0.0]] [] 0 true true false false -0.3586117992626559 0 0.858760484549 ["Character/Dirichlet/145/103"] "1-145-145.104-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.104" [[1, 0.0]] [] 0 true true false false -0.030279735397710852 0 0.253433008145 ["Character/Dirichlet/145/104"] "1-145-145.107-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.107" [[1, 0.0]] [] 0 true true false false 0.3586117992626559 0 2.27834028694 ["Character/Dirichlet/145/107"] "1-145-145.112-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.112" [[1, 0.0]] [] 0 true true false false -0.05413545118617167 0 1.32016383718 ["Character/Dirichlet/145/112"] "1-145-145.114-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.114" [[1, 0.0]] [] 0 true true false false 0.24388423879611432 0 2.04692817851 ["Character/Dirichlet/145/114"] "1-145-145.119-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.119" [[1, 0.0]] [] 0 true true false false 0.05161134613583512 0 0.821961425648 ["Character/Dirichlet/145/119"] "1-145-145.122-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.122" [[1, 0.0]] [] 0 true true false false 0.49424633873753 0 0.267603078172 ["Character/Dirichlet/145/122"] "1-145-145.123-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.123" [[1, 0.0]] [] 0 true true false false 0.05413545118617167 0 0.607593968063 ["Character/Dirichlet/145/123"] "1-145-145.124-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.124" [[1, 0.0]] [] 0 true true false false -0.2806960991935661 0 0.687819641167 ["Character/Dirichlet/145/124"] "1-145-145.13-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.13" [[1, 0.0]] [] 0 true true false false -0.41083655267840974 0 1.45760362165 ["Character/Dirichlet/145/13"] "1-145-145.132-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.132" [[1, 0.0]] [] 0 true true false false -0.18240360808787254 0 0.323028501874 ["Character/Dirichlet/145/132"] "1-145-145.134-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.134" [[1, 0.0]] [] 0 true true false false -0.47255262805799003 0 0.330244367268 ["Character/Dirichlet/145/134"] "1-145-145.138-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.138" [[1, 0.0]] [] 0 true true false false -0.316978795828548 0 1.00068594303 ["Character/Dirichlet/145/138"] "1-145-145.14-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.14" [[1, 0.0]] [] 0 true true false false -0.24388423879611432 0 1.22113163001 ["Character/Dirichlet/145/14"] "1-145-145.19-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.19" [[1, 0.0]] [] 0 true true false false 0.08602174618767422 0 1.73360626992 ["Character/Dirichlet/145/19"] "1-145-145.22-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.22" [[1, 0.0]] [] 0 true true false false -0.14077060465376462 0 2.95909275192 ["Character/Dirichlet/145/22"] "1-145-145.23-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.23" [[1, 0.0]] [] 0 true true false false -0.28057822053539516 0 0.574569809257 ["Character/Dirichlet/145/23"] "1-145-145.28-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.28" [[1, 0.0]] [] 0 true true false false 0.4118959044126083 0 1.42149042416 ["Character/Dirichlet/145/28"] "1-145-145.33-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.33" [[1, 0.0]] [] 0 true true false false 0.14077060465376462 0 0.00430465762665 ["Character/Dirichlet/145/33"] "1-145-145.38-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.38" [[1, 0.0]] [] 0 true true false false 0.23462836150362637 0 1.96463587435 ["Character/Dirichlet/145/38"] "1-145-145.39-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.39" [[1, 0.0]] [] 0 true true false false -0.05161134613583512 0 0.41310323539 ["Character/Dirichlet/145/39"] "1-145-145.42-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.42" [[1, 0.0]] [] 0 true true false false -0.23462836150362637 0 1.39291410554 ["Character/Dirichlet/145/42"] "1-145-145.44-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.44" [[1, 0.0]] [] 0 true true false false 0.08643810998352944 0 0.827576870109 ["Character/Dirichlet/145/44"] "1-145-145.52-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.52" [[1, 0.0]] [] 0 true true false false -0.1043700293606118 0 0.9733179846 ["Character/Dirichlet/145/52"] "1-145-145.53-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.53" [[1, 0.0]] [] 0 true true false false 0.1043700293606118 0 1.65075013258 ["Character/Dirichlet/145/53"] "1-145-145.57-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.57" [[1, 0.0]] [] 0 true true false false -0.4118959044126083 0 0.193091479238 ["Character/Dirichlet/145/57"] "1-145-145.62-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.62" [[1, 0.0]] [] 0 true true false false 0.316978795828548 0 1.45407154242 ["Character/Dirichlet/145/62"] "1-145-145.63-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.63" [[1, 0.0]] [] 0 true true false false -0.49424633873753 0 2.07052955006 ["Character/Dirichlet/145/63"] "1-145-145.67-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.67" [[1, 0.0]] [] 0 true true false false 0.41083655267840974 0 0.0910806605233 ["Character/Dirichlet/145/67"] "1-145-145.69-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.69" [[1, 0.0]] [] 0 true true false false 0.2806960991935661 0 1.59714566769 ["Character/Dirichlet/145/69"] "1-145-145.7-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.7" [[1, 0.0]] [] 0 true true false false 0.23034364236095503 0 1.70652967 ["Character/Dirichlet/145/7"] "1-145-145.78-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.78" [[1, 0.0]] [] 0 true true false false 0.18240360808787254 0 1.19939641398 ["Character/Dirichlet/145/78"] "1-145-145.79-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.79" [[1, 0.0]] [] 0 true true false false 0.47255262805799003 0 1.87934297646 ["Character/Dirichlet/145/79"] "1-145-145.82-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.82" [[1, 0.0]] [] 0 true true false false 0.28057822053539516 0 1.4199018038 ["Character/Dirichlet/145/82"] "1-145-145.83-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.83" [[1, 0.0]] [] 0 true true false false -0.23034364236095503 0 0.0961958833057 ["Character/Dirichlet/145/83"] "1-145-145.84-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.84" [[1, 0.0]] [] 0 true true false false -0.08602174618767422 0 0.762219651057 ["Character/Dirichlet/145/84"] "1-145-145.89-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.89" [[1, 0.0]] [] 0 true true false false -0.08643810998352944 0 1.06655964562 ["Character/Dirichlet/145/89"] "1-145-145.92-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.92" [[1, 0.0]] [] 0 true true false false -0.3180381475627466 0 0.597511800227 ["Character/Dirichlet/145/92"] "1-145-145.93-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.93" [[1, 0.0]] [] 0 true true false false 0.3180381475627466 0 2.05184084001 ["Character/Dirichlet/145/93"] "1-145-145.99-r1-0-0" 15.582414095220628 15.582414095220628 1 145 "145.99" [[1, 0.0]] [] 0 true true false false 0.030279735397710852 0 1.29694840521 ["Character/Dirichlet/145/99"] # Label -- # Each L-function $L$ has a label of the form d-N-q.k-x-y-i, where # * $d$ is the degree of $L$. # * $N$ is the conductor of $L$. When $N$ is a perfect power $m^n$ we write $N$ as $m$e$n$, since $N$ can be very large for some imprimitive L-functions. # * q.k is the label of the primitive Dirichlet character from which the central character is induced. # * x-y is the spectral label encoding the $\mu_j$ and $\nu_j$ in the analytically normalized functional equation. # * i is a non-negative integer disambiguating between L-functions that would otherwise have the same label. #$\alpha$ (root_analytic_conductor) -- # If $d$ is the degree of the L-function $L(s)$, the **root analytic conductor** $\alpha$ of $L$ is the $d$th root of the analytic conductor of $L$. It plays a role analogous to the root discriminant for number fields. #$A$ (analytic_conductor) -- # The **analytic conductor** of an L-function $L(s)$ with infinity factor $L_{\infty}(s)$ and conductor $N$ is the real number # \[ # A := \mathrm{exp}\left(2\mathrm{Re}\left(\frac{L_{\infty}'(1/2)}{L_{\infty}(1/2)}\right)\right)N. # \] #$d$ (degree) -- # The **degree** of an L-function is the number $J + 2K$ of Gamma factors occurring in its functional equation # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). # \] # The degree appears as the first component of the Selberg data of $L(s).$ In all known cases it is the degree of the polynomial of the inverse of the Euler factor at any prime not dividing the conductor. #$N$ (conductor) -- # The **conductor** of an L-function is the integer $N$ occurring in its functional equation # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). # \] # The conductor of an analytic L-function is the second component in the Selberg data. For a Dirichlet L-function # associated with a primitive Dirichlet character, the conductor of the L-function is the same as the conductor of the character. For a primitive L-function associated with a cusp form $\phi$ on $GL(2)/\mathbb Q$, the conductor of the L-function is the same as the level of $\phi$. # In the literature, the word _level_ is sometimes used instead of _conductor_. #$\chi$ (central_character) -- # An L-function has an Euler product of the form # $L(s) = \prod_p L_p(p^{-s})^{-1}$ # where $L_p(x) = 1 + a_p x + \ldots + (-1)^d \chi(p) x^d$. The character $\chi$ is a Dirichlet character mod $N$ and is called **central character** of the L-function. # Here, $N$ is the conductor of $L$. #$\mu$ (mus) -- # All known analytic L-functions have a **functional equation** that can be written in the form # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), # \] # where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer # or half-integer, # \[ # \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, # \] # and $\varepsilon$ is the sign of the functional equation. # With those restrictions on the spectral parameters, the # data in the functional equation is specified uniquely. The integer $d = J + 2 K$ # is the degree of the L-function. The integer $N$ is the conductor (or level) # of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters # in the functional equation can be used to make up the 4-tuple called the Selberg data. # The axioms of the Selberg class are less restrictive than # given above. # Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$. # For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, # called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers. #$\nu$ (nus) -- # All known analytic L-functions have a **functional equation** that can be written in the form # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), # \] # where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer # or half-integer, # \[ # \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, # \] # and $\varepsilon$ is the sign of the functional equation. # With those restrictions on the spectral parameters, the # data in the functional equation is specified uniquely. The integer $d = J + 2 K$ # is the degree of the L-function. The integer $N$ is the conductor (or level) # of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters # in the functional equation can be used to make up the 4-tuple called the Selberg data. # The axioms of the Selberg class are less restrictive than # given above. # Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$. # For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, # called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers. #$w$ (motivic_weight) -- # The **motivic weight** (or **arithmetic weight**) of an arithmetic L-function with analytic normalization $L_{an}(s)=\sum_{n=1}^\infty a_nn^{-s}$ is the least nonnegative integer $w$ for which $a_nn^{w/2}$ is an algebraic integer for all $n\ge 1$. # If the L-function arises from a motive, then the weight of the motive has the # same parity as the motivic weight of the L-function, but the weight of the motive # could be larger. This apparent discrepancy comes from the fact that a Tate twist # increases the weight of the motive. This corresponds to the change of variables # $s \mapsto s + j$ in the L-function of the motive. #prim (primitive) -- # An L-function is primitive if it cannot be written as a product of nontrivial L-functions. The "trivial L-function" is the constant function $1$. #arith (algebraic) -- # An L-function $L(s) = \sum_{n=1}^{\infty} a_n n^{-s}$ is called **arithmetic** if its Dirichlet coefficients $a_n$ are algebraic numbers. #$\mathbb{Q}$ (rational) -- # A **rational** L-function $L(s)$ is an arithmetic L-function with coefficient field $\Q$; equivalently, its Euler product in the arithmetic normalization can be written as a product over rational primes # \[ # L(s)=\prod_pL_p(p^{-s})^{-1} # \] # with $L_p\in \Z[T]$. #self-dual (self_dual) -- # An L-function $L(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ is called **self-dual** if its Dirichlet coefficients $a_n$ are real. #$\operatorname{Arg}(\epsilon)$ (root_angle) -- # The **root angle** of an L-function is the argument of its root number, as a real number $\alpha$ with $-0.5 < \alpha \le 0.5$. #$r$ (order_of_vanishing) -- # The **analytic rank** of an L-function $L(s)$ is its order of vanishing at its central point. # When the analytic rank $r$ is positive, the value listed in the LMFDB is typically an upper bound that is believed to be tight (in the sense that there are known to be $r$ zeroes located very near to the central point). #First zero (z1) -- # The **zeros** of an L-function $L(s)$ are the complex numbers $\rho$ for which $L(\rho)=0$. # Under the Riemann Hypothesis, every non-trivial zero $\rho$ lies on the critical line $\Re(s)=1/2$ (in the analytic normalization). # The **lowest zero** of an L-function $L(s)$ is the least $\gamma>0$ for which $L(1/2+i\gamma)=0$. Note that even when $L(1/2)=0$, the lowest zero is by definition a positive real number. #Origin (instance_urls) -- # L-functions arise from many different sources. Already in degree 2 we have examples of # L-functions associated with holomorphic cusp forms, with Maass forms, with elliptic curves, with characters of number fields (Hecke characters), and with 2-dimensional representations of the Galois group of a number field (Artin L-functions). # Sometimes an L-function may arise from more than one source. For example, the L-functions associated with elliptic curves are also associated with weight 2 cusp forms. A goal of the Langlands program ostensibly is to prove that any degree $d$ L-function is associated with an automorphic form on $\mathrm{GL}(d)$. Because of this representation theoretic genesis, one can associate an L-function not only to an automorphic representation but also to symmetric powers, or exterior powers of that representation, or to the tensor product of two representations (the Rankin-Selberg product of two L-functions).