# lfunc_search downloaded from the LMFDB on 25 June 2026. # Search link: https://www.lmfdb.org/L/1/21^2/441.232/r0-0 # Query "{'degree': 1, 'conductor': 441, 'spectral_label': 'r0-0'}" returned 72 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. "1-21e2-441.101-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.101" [[0, 0.0]] [] 0 true true false false -0.20975056689342403 0 0.922390337765 ["Character/Dirichlet/441/101"] "1-21e2-441.104-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.104" [[0, 0.0]] [] 0 true true false false -0.45691609977324266 0 0.328335827956 ["Character/Dirichlet/441/104"] "1-21e2-441.106-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.106" [[0, 0.0]] [] 0 true true false false -0.04308390022675738 0 1.43511919454 ["Character/Dirichlet/441/106"] "1-21e2-441.110-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.110" [[0, 0.0]] [] 0 true true false false -0.05328798185941043 0 1.18832358395 ["Character/Dirichlet/441/110"] "1-21e2-441.121-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.121" [[0, 0.0]] [] 0 true true false false -0.4467120181405896 0 0.664213625683 ["Character/Dirichlet/441/121"] "1-21e2-441.122-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.122" [[0, 0.0]] [] 0 true true false false 0.4852607709750567 0 0.498567357998 ["Character/Dirichlet/441/122"] "1-21e2-441.130-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.130" [[0, 0.0]] [] 0 true true false false 0.0022675736961451248 0 1.2640276551 ["Character/Dirichlet/441/130"] "1-21e2-441.131-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.131" [[0, 0.0]] [] 0 true true false false 0.20975056689342403 0 0.971372047053 ["Character/Dirichlet/441/131"] "1-21e2-441.142-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.142" [[0, 0.0]] [] 0 true true false false -0.3764172335600907 0 0.54320827123 ["Character/Dirichlet/441/142"] "1-21e2-441.151-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.151" [[0, 0.0]] [] 0 true true false false 0.0022675736961451248 0 1.5326766282 ["Character/Dirichlet/441/151"] "1-21e2-441.16-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.16" [[0, 0.0]] [] 0 true true false false 0.1541950113378685 0 1.72225835702 ["Character/Dirichlet/441/16"] "1-21e2-441.164-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.164" [[0, 0.0]] [] 0 true true false false -0.05328798185941043 0 1.21623232814 ["Character/Dirichlet/441/164"] "1-21e2-441.167-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.167" [[0, 0.0]] [] 0 true true false false 0.0022675736961451248 0 1.66359342738 ["Character/Dirichlet/441/167"] "1-21e2-441.169-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.169" [[0, 0.0]] [] 0 true true false false 0.0022675736961451248 0 1.13601416493 ["Character/Dirichlet/441/169"] "1-21e2-441.173-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.173" [[0, 0.0]] [] 0 true true false false 0.0022675736961451248 0 1.0269921195 ["Character/Dirichlet/441/173"] "1-21e2-441.184-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.184" [[0, 0.0]] [] 0 true true false false 0.0022675736961451248 0 0.890593094116 ["Character/Dirichlet/441/184"] "1-21e2-441.185-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.185" [[0, 0.0]] [] 0 true true false false -0.09637188208616779 0 0.813329686714 ["Character/Dirichlet/441/185"] "1-21e2-441.193-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.193" [[0, 0.0]] [] 0 true true false false -0.1541950113378685 0 1.3668535709 ["Character/Dirichlet/441/193"] "1-21e2-441.194-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.194" [[0, 0.0]] [] 0 true true false false -0.4297052154195012 0 0.551595208637 ["Character/Dirichlet/441/194"] "1-21e2-441.20-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.20" [[0, 0.0]] [] 0 true true false false -0.05328798185941043 0 0.690870007214 ["Character/Dirichlet/441/20"] "1-21e2-441.205-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.205" [[0, 0.0]] [] 0 true true false false -0.4036281179138322 0 0.757534618603 ["Character/Dirichlet/441/205"] "1-21e2-441.209-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.209" [[0, 0.0]] [] 0 true true false false 0.18140589569161 0 1.21748387954 ["Character/Dirichlet/441/209"] "1-21e2-441.211-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.211" [[0, 0.0]] [] 0 true true false false -0.18140589569161 0 1.03060564945 ["Character/Dirichlet/441/211"] "1-21e2-441.22-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.22" [[0, 0.0]] [] 0 true true false false -0.4467120181405896 0 2.09262158218 ["Character/Dirichlet/441/22"] "1-21e2-441.230-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.230" [[0, 0.0]] [] 0 true true false false -0.18140589569161 0 0.299588008106 ["Character/Dirichlet/441/230"] "1-21e2-441.232-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.232" [[0, 0.0]] [] 0 true true false false 0.18140589569161 0 1.38826369413 ["Character/Dirichlet/441/232"] "1-21e2-441.236-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.236" [[0, 0.0]] [] 0 true true false false 0.09637188208616779 0 1.06285790282 ["Character/Dirichlet/441/236"] "1-21e2-441.247-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.247" [[0, 0.0]] [] 0 true true false false 0.07029478458049887 0 0.961195760347 ["Character/Dirichlet/441/247"] "1-21e2-441.248-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.248" [[0, 0.0]] [] 0 true true false false -0.1541950113378685 0 0.776034816071 ["Character/Dirichlet/441/248"] "1-21e2-441.25-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.25" [[0, 0.0]] [] 0 true true false false -0.07029478458049887 0 0.806939185452 ["Character/Dirichlet/441/25"] "1-21e2-441.256-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.256" [[0, 0.0]] [] 0 true true false false 0.4036281179138322 0 2.13659127607 ["Character/Dirichlet/441/256"] "1-21e2-441.257-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.257" [[0, 0.0]] [] 0 true true false false 0.0022675736961451248 0 1.26113580945 ["Character/Dirichlet/441/257"] "1-21e2-441.268-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.268" [[0, 0.0]] [] 0 true true false false 0.0022675736961451248 0 0.611475122906 ["Character/Dirichlet/441/268"] "1-21e2-441.272-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.272" [[0, 0.0]] [] 0 true true false false 0.0022675736961451248 0 1.51568788839 ["Character/Dirichlet/441/272"] "1-21e2-441.274-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.274" [[0, 0.0]] [] 0 true true false false 0.0022675736961451248 0 0.945089397682 ["Character/Dirichlet/441/274"] "1-21e2-441.277-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.277" [[0, 0.0]] [] 0 true true false false 0.4467120181405896 0 2.15155888262 ["Character/Dirichlet/441/277"] "1-21e2-441.290-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.290" [[0, 0.0]] [] 0 true true false false 0.0022675736961451248 0 0.655964960072 ["Character/Dirichlet/441/290"] "1-21e2-441.299-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.299" [[0, 0.0]] [] 0 true true false false 0.1235827664399093 0 1.330250036 ["Character/Dirichlet/441/299"] "1-21e2-441.310-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.310" [[0, 0.0]] [] 0 true true false false -0.29024943310657597 0 0.546957046619 ["Character/Dirichlet/441/310"] "1-21e2-441.311-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.311" [[0, 0.0]] [] 0 true true false false 0.0022675736961451248 0 0.50295989047 ["Character/Dirichlet/441/311"] "1-21e2-441.319-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.319" [[0, 0.0]] [] 0 true true false false 0.4852607709750567 0 1.50543217126 ["Character/Dirichlet/441/319"] "1-21e2-441.320-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.320" [[0, 0.0]] [] 0 true true false false 0.05328798185941043 0 1.07745128942 ["Character/Dirichlet/441/320"] "1-21e2-441.331-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.331" [[0, 0.0]] [] 0 true true false false 0.4467120181405896 0 1.55259665311 ["Character/Dirichlet/441/331"] "1-21e2-441.335-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.335" [[0, 0.0]] [] 0 true true false false 0.45691609977324266 0 1.82263669353 ["Character/Dirichlet/441/335"] "1-21e2-441.337-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.337" [[0, 0.0]] [] 0 true true false false 0.04308390022675738 0 1.4666422559 ["Character/Dirichlet/441/337"] "1-21e2-441.340-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.340" [[0, 0.0]] [] 0 true true false false 0.29024943310657597 0 0.992598558942 ["Character/Dirichlet/441/340"] "1-21e2-441.353-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.353" [[0, 0.0]] [] 0 true true false false -0.17913832199546487 0 0.36950104295 ["Character/Dirichlet/441/353"] "1-21e2-441.356-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.356" [[0, 0.0]] [] 0 true true false false 0.23696145124716556 0 1.2414291239 ["Character/Dirichlet/441/356"] "1-21e2-441.358-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.358" [[0, 0.0]] [] 0 true true false false 0.2630385487528345 0 1.32529244594 ["Character/Dirichlet/441/358"] "1-21e2-441.38-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.38" [[0, 0.0]] [] 0 true true false false -0.15192743764172337 0 1.39431793628 ["Character/Dirichlet/441/38"] "1-21e2-441.382-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.382" [[0, 0.0]] [] 0 true true false false 0.3764172335600907 0 1.53294647413 ["Character/Dirichlet/441/382"] "1-21e2-441.383-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.383" [[0, 0.0]] [] 0 true true false false 0.15192743764172337 0 1.64476225983 ["Character/Dirichlet/441/383"] "1-21e2-441.394-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.394" [[0, 0.0]] [] 0 true true false false -0.4852607709750567 0 0.265091215006 ["Character/Dirichlet/441/394"] "1-21e2-441.398-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.398" [[0, 0.0]] [] 0 true true false false 0.4875283446712019 0 0.368176345946 ["Character/Dirichlet/441/398"] "1-21e2-441.4-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.4" [[0, 0.0]] [] 0 true true false false -0.4467120181405896 0 0.226478204666 ["Character/Dirichlet/441/4"] "1-21e2-441.400-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.400" [[0, 0.0]] [] 0 true true false false -0.4875283446712019 0 0.228606566072 ["Character/Dirichlet/441/400"] "1-21e2-441.403-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.403" [[0, 0.0]] [] 0 true true false false -0.15192743764172337 0 0.727721358751 ["Character/Dirichlet/441/403"] "1-21e2-441.41-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.41" [[0, 0.0]] [] 0 true true false false -0.4875283446712019 0 2.08013614603 ["Character/Dirichlet/441/41"] "1-21e2-441.416-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.416" [[0, 0.0]] [] 0 true true false false 0.4297052154195012 0 2.25235784194 ["Character/Dirichlet/441/416"] "1-21e2-441.419-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.419" [[0, 0.0]] [] 0 true true false false 0.05328798185941043 0 1.28124289699 ["Character/Dirichlet/441/419"] "1-21e2-441.421-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.421" [[0, 0.0]] [] 0 true true false false 0.4467120181405896 0 0.516458658469 ["Character/Dirichlet/441/421"] "1-21e2-441.425-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.425" [[0, 0.0]] [] 0 true true false false 0.1541950113378685 0 1.37368538365 ["Character/Dirichlet/441/425"] "1-21e2-441.43-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.43" [[0, 0.0]] [] 0 true true false false 0.4875283446712019 0 1.44742811439 ["Character/Dirichlet/441/43"] "1-21e2-441.436-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.436" [[0, 0.0]] [] 0 true true false false 0.17913832199546487 0 1.03366962719 ["Character/Dirichlet/441/436"] "1-21e2-441.437-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.437" [[0, 0.0]] [] 0 true true false false 0.05328798185941043 0 1.34845253506 ["Character/Dirichlet/441/437"] "1-21e2-441.47-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.47" [[0, 0.0]] [] 0 true true false false -0.4852607709750567 0 1.67909390091 ["Character/Dirichlet/441/47"] "1-21e2-441.5-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.5" [[0, 0.0]] [] 0 true true false false 0.17913832199546487 0 1.73238126981 ["Character/Dirichlet/441/5"] "1-21e2-441.58-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.58" [[0, 0.0]] [] 0 true true false false 0.15192743764172337 0 1.58205223918 ["Character/Dirichlet/441/58"] "1-21e2-441.59-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.59" [[0, 0.0]] [] 0 true true false false -0.1235827664399093 0 1.34429523494 ["Character/Dirichlet/441/59"] "1-21e2-441.83-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.83" [[0, 0.0]] [] 0 true true false false -0.23696145124716556 0 0.933668113512 ["Character/Dirichlet/441/83"] "1-21e2-441.85-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.85" [[0, 0.0]] [] 0 true true false false -0.2630385487528345 0 1.06492209075 ["Character/Dirichlet/441/85"] "1-21e2-441.88-r0-0-0" 2.047995390047135 2.047995390047135 1 441 "441.88" [[0, 0.0]] [] 0 true true false false -0.17913832199546487 0 1.21472431802 ["Character/Dirichlet/441/88"] # 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).