# lfunc_search downloaded from the LMFDB on 14 June 2026. # Search link: https://www.lmfdb.org/L/2/3675/105.17/c0-0 # Query "{'degree': 2, 'conductor': 3675, 'spectral_label': 'c0-0'}" returned 292 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. "2-3675-105.17-c0-0-0" 1.3542761614332153 1.834063921426284 2 3675 "105.17" [] [[0.0, 0.0]] 0 true true false false 0.48991343165138845 0 0.14971668893745132076026012694 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/d/2432/1"] "2-3675-105.17-c0-0-1" 1.3542761614332153 1.834063921426284 2 3675 "105.17" [] [[0.0, 0.0]] 0 true true false false -0.2654782918891393 0 0.77476946922978747594034603304 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/c/2432/3"] "2-3675-105.17-c0-0-10" 1.3542761614332153 1.834063921426284 2 3675 "105.17" [] [[0.0, 0.0]] 0 true true false false 0.2809588267273377 0 1.61503739749990397508887998196 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/b/2432/2"] "2-3675-105.17-c0-0-11" 1.3542761614332153 1.834063921426284 2 3675 "105.17" [] [[0.0, 0.0]] 0 true true false false -0.4233787315225046 0 2.17177903320642383545653296540 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/d/2432/3"] "2-3675-105.17-c0-0-2" 1.3542761614332153 1.834063921426284 2 3675 "105.17" [] [[0.0, 0.0]] 0 true true false false 0.19785368772254272 0 0.806584499105696930235848655601 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/c/2432/1"] "2-3675-105.17-c0-0-3" 1.3542761614332153 1.834063921426284 2 3675 "105.17" [] [[0.0, 0.0]] 0 true true false false 0.06300593515182511 0 0.846542425733364151394629196099 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/b/2432/1"] "2-3675-105.17-c0-0-4" 1.3542761614332153 1.834063921426284 2 3675 "105.17" [] [[0.0, 0.0]] 0 true true false false 0.23452170811086076 0 0.865166304301552755454089561075 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/d/2432/2"] "2-3675-105.17-c0-0-5" 1.3542761614332153 1.834063921426284 2 3675 "105.17" [] [[0.0, 0.0]] 0 true true false false 0.05652390544769808 0 1.12749089989847682664994112249 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/a/2432/2"] "2-3675-105.17-c0-0-6" 1.3542761614332153 1.834063921426284 2 3675 "105.17" [] [[0.0, 0.0]] 0 true true false false 0.07662126847749537 0 1.12923670489695540489361418254 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/c/2432/2"] "2-3675-105.17-c0-0-7" 1.3542761614332153 1.834063921426284 2 3675 "105.17" [] [[0.0, 0.0]] 0 true true false false -0.3021463122774573 0 1.30368545698763321955413762796 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/d/2432/4"] "2-3675-105.17-c0-0-8" 1.3542761614332153 1.834063921426284 2 3675 "105.17" [] [[0.0, 0.0]] 0 true true false false -0.010086568348611548 0 1.49518164657277121840355935585 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/c/2432/4"] "2-3675-105.17-c0-0-9" 1.3542761614332153 1.834063921426284 2 3675 "105.17" [] [[0.0, 0.0]] 0 true true false false -0.44347609455230197 0 1.60225585707809007510935911521 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/a/2432/1"] "2-3675-105.38-c0-0-0" 1.3542761614332153 1.834063921426284 2 3675 "105.38" [] [[0.0, 0.0]] 0 true true false false -0.36968428572708534 0 0.72520642606654638789519199413 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/a/668/2"] "2-3675-105.38-c0-0-1" 1.3542761614332153 1.834063921426284 2 3675 "105.38" [] [[0.0, 0.0]] 0 true true false false -0.349586922697288 0 0.72567782353299144931319037460 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/c/668/2"] "2-3675-105.38-c0-0-10" 1.3542761614332153 1.834063921426284 2 3675 "105.38" [] [[0.0, 0.0]] 0 true true false false 0.3083135169360774 0 1.52778079010485811923395335398 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/c/668/3"] "2-3675-105.38-c0-0-11" 1.3542761614332153 1.834063921426284 2 3675 "105.38" [] [[0.0, 0.0]] 0 true true false false -0.4362947595233949 0 1.88962225822816735616961867997 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/c/668/4"] "2-3675-105.38-c0-0-2" 1.3542761614332153 1.834063921426284 2 3675 "105.38" [] [[0.0, 0.0]] 0 true true false false 0.2716454965477594 0 0.76342559728662859154927675580 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/d/668/4"] "2-3675-105.38-c0-0-3" 1.3542761614332153 1.834063921426284 2 3675 "105.38" [] [[0.0, 0.0]] 0 true true false false -0.04283298209787899 0 0.850481352622514594211939691985 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/b/668/1"] "2-3675-105.38-c0-0-4" 1.3542761614332153 1.834063921426284 2 3675 "105.38" [] [[0.0, 0.0]] 0 true true false false -0.2607858736733915 0 1.01146188581131703457315116844 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/b/668/2"] "2-3675-105.38-c0-0-5" 1.3542761614332153 1.834063921426284 2 3675 "105.38" [] [[0.0, 0.0]] 0 true true false false -0.22835450345224065 0 1.15456252051094290377494477854 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/c/668/1"] "2-3675-105.38-c0-0-6" 1.3542761614332153 1.834063921426284 2 3675 "105.38" [] [[0.0, 0.0]] 0 true true false false -0.19168648306392264 0 1.16530306750135656113068198507 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/d/668/2"] "2-3675-105.38-c0-0-7" 1.3542761614332153 1.834063921426284 2 3675 "105.38" [] [[0.0, 0.0]] 0 true true false false 0.0637052404766051 0 1.24916496656102105247568696529 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/d/668/1"] "2-3675-105.38-c0-0-8" 1.3542761614332153 1.834063921426284 2 3675 "105.38" [] [[0.0, 0.0]] 0 true true false false 0.13031571427291472 0 1.28526090314831825404380804987 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/a/668/1"] "2-3675-105.38-c0-0-9" 1.3542761614332153 1.834063921426284 2 3675 "105.38" [] [[0.0, 0.0]] 0 true true false false 0.150413077302712 0 1.48958428070256291146154484119 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/d/668/3"] "2-3675-105.44-c0-0-0" 1.3542761614332153 1.834063921426284 2 3675 "105.44" [] [[0.0, 0.0]] 0 true true false false 0.3961004527072645 0 0.16588293812962109539062318672 ["ModularForm/GL2/Q/holomorphic/3675/1/p/b/2774/1"] "2-3675-105.44-c0-0-1" 1.3542761614332153 1.834063921426284 2 3675 "105.44" [] [[0.0, 0.0]] 0 true true false false -0.4805236900783106 0 0.61531571448270503956895222268 ["ModularForm/GL2/Q/holomorphic/3675/1/p/b/2774/4"] "2-3675-105.44-c0-0-2" 1.3542761614332153 1.834063921426284 2 3675 "105.44" [] [[0.0, 0.0]] 0 true true false false -0.32270190478750393 0 0.66246478570059125016810864412 ["ModularForm/GL2/Q/holomorphic/3675/1/p/a/2774/1"] "2-3675-105.44-c0-0-3" 1.3542761614332153 1.834063921426284 2 3675 "105.44" [] [[0.0, 0.0]] 0 true true false false -0.20631592964230225 0 0.76217796079752179463253421616 ["ModularForm/GL2/Q/holomorphic/3675/1/p/b/2774/3"] "2-3675-105.44-c0-0-4" 1.3542761614332153 1.834063921426284 2 3675 "105.44" [] [[0.0, 0.0]] 0 true true false false 0.16705992757212274 0 1.04837713854526043410803941527 ["ModularForm/GL2/Q/holomorphic/3675/1/p/c/2774/1"] "2-3675-105.44-c0-0-5" 1.3542761614332153 1.834063921426284 2 3675 "105.44" [] [[0.0, 0.0]] 0 true true false false 0.04368407035769777 0 1.27496347435073288134158252586 ["ModularForm/GL2/Q/holomorphic/3675/1/p/c/2774/4"] "2-3675-105.44-c0-0-6" 1.3542761614332153 1.834063921426284 2 3675 "105.44" [] [[0.0, 0.0]] 0 true true false false -0.08294007242787729 0 1.30387791676171214020147488210 ["ModularForm/GL2/Q/holomorphic/3675/1/p/b/2774/2"] "2-3675-105.44-c0-0-7" 1.3542761614332153 1.834063921426284 2 3675 "105.44" [] [[0.0, 0.0]] 0 true true false false 0.2694763099216894 0 1.40478001582320235909862809332 ["ModularForm/GL2/Q/holomorphic/3675/1/p/c/2774/3"] "2-3675-105.44-c0-0-8" 1.3542761614332153 1.834063921426284 2 3675 "105.44" [] [[0.0, 0.0]] 0 true true false false 0.029714477562062823 0 1.41504440321958248405963479284 ["ModularForm/GL2/Q/holomorphic/3675/1/p/a/2774/2"] "2-3675-105.44-c0-0-9" 1.3542761614332153 1.834063921426284 2 3675 "105.44" [] [[0.0, 0.0]] 0 true true false false 0.14610045270726452 0 1.58705141984520713798915670032 ["ModularForm/GL2/Q/holomorphic/3675/1/p/c/2774/2"] "2-3675-105.47-c0-0-0" 1.3542761614332153 1.834063921426284 2 3675 "105.47" [] [[0.0, 0.0]] 0 true true false false -0.0637052404766051 0 0.10568152572887587566530365128 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/d/1832/1"] "2-3675-105.47-c0-0-1" 1.3542761614332153 1.834063921426284 2 3675 "105.47" [] [[0.0, 0.0]] 0 true true false false -0.3083135169360774 0 0.61801979358314675053550827620 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/c/1832/3"] "2-3675-105.47-c0-0-10" 1.3542761614332153 1.834063921426284 2 3675 "105.47" [] [[0.0, 0.0]] 0 true true false false 0.2607858736733915 0 1.62066041008479221296894685630 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/b/1832/2"] "2-3675-105.47-c0-0-11" 1.3542761614332153 1.834063921426284 2 3675 "105.47" [] [[0.0, 0.0]] 0 true true false false 0.36968428572708534 0 1.96431044050381027291904265876 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/a/1832/2"] "2-3675-105.47-c0-0-2" 1.3542761614332153 1.834063921426284 2 3675 "105.47" [] [[0.0, 0.0]] 0 true true false false 0.4362947595233949 0 0.840052767228404580133995427324 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/c/1832/4"] "2-3675-105.47-c0-0-3" 1.3542761614332153 1.834063921426284 2 3675 "105.47" [] [[0.0, 0.0]] 0 true true false false 0.22835450345224065 0 0.964415023536148615930712701650 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/c/1832/1"] "2-3675-105.47-c0-0-4" 1.3542761614332153 1.834063921426284 2 3675 "105.47" [] [[0.0, 0.0]] 0 true true false false -0.2716454965477594 0 1.02361241859950335335140153982 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/d/1832/4"] "2-3675-105.47-c0-0-5" 1.3542761614332153 1.834063921426284 2 3675 "105.47" [] [[0.0, 0.0]] 0 true true false false -0.13031571427291472 0 1.04029045000882056003321235374 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/a/1832/1"] "2-3675-105.47-c0-0-6" 1.3542761614332153 1.834063921426284 2 3675 "105.47" [] [[0.0, 0.0]] 0 true true false false 0.349586922697288 0 1.22730733561502072127776208038 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/c/1832/2"] "2-3675-105.47-c0-0-7" 1.3542761614332153 1.834063921426284 2 3675 "105.47" [] [[0.0, 0.0]] 0 true true false false 0.04283298209787899 0 1.35661043091872302330523056939 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/b/1832/1"] "2-3675-105.47-c0-0-8" 1.3542761614332153 1.834063921426284 2 3675 "105.47" [] [[0.0, 0.0]] 0 true true false false 0.19168648306392264 0 1.46489094962983220048873328407 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/d/1832/2"] "2-3675-105.47-c0-0-9" 1.3542761614332153 1.834063921426284 2 3675 "105.47" [] [[0.0, 0.0]] 0 true true false false -0.150413077302712 0 1.56485842481918826389714263655 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/d/1832/3"] "2-3675-105.62-c0-0-0" 1.3542761614332153 1.834063921426284 2 3675 "105.62" [] [[0.0, 0.0]] 0 true true false false -0.35397555656034857 0 0.69535826608653726267821267919 ["ModularForm/GL2/Q/holomorphic/3675/1/k/a/2057/2"] "2-3675-105.62-c0-0-1" 1.3542761614332153 1.834063921426284 2 3675 "105.62" [] [[0.0, 0.0]] 0 true true false false -0.21981625226486814 0 0.71248444538622746935575364005 ["ModularForm/GL2/Q/holomorphic/3675/1/k/a/2057/3"] "2-3675-105.62-c0-0-10" 1.3542761614332153 1.834063921426284 2 3675 "105.62" [] [[0.0, 0.0]] 0 true true false false 0.2801837477351319 0 1.79270892847204493427848282850 ["ModularForm/GL2/Q/holomorphic/3675/1/k/c/2057/2"] "2-3675-105.62-c0-0-11" 1.3542761614332153 1.834063921426284 2 3675 "105.62" [] [[0.0, 0.0]] 0 true true false false 0.4881240038062862 0 2.31938760150733969190746355312 ["ModularForm/GL2/Q/holomorphic/3675/1/k/a/2057/4"] "2-3675-105.62-c0-0-2" 1.3542761614332153 1.834063921426284 2 3675 "105.62" [] [[0.0, 0.0]] 0 true true false false -0.09272041271388794 0 0.73440946939596722750165792151 ["ModularForm/GL2/Q/holomorphic/3675/1/k/b/2057/2"] "2-3675-105.62-c0-0-3" 1.3542761614332153 1.834063921426284 2 3675 "105.62" [] [[0.0, 0.0]] 0 true true false false -0.36553488688538294 0 0.76506401512182808837117888086 ["ModularForm/GL2/Q/holomorphic/3675/1/k/b/2057/3"] "2-3675-105.62-c0-0-4" 1.3542761614332153 1.834063921426284 2 3675 "105.62" [] [[0.0, 0.0]] 0 true true false false -0.3106733042894005 0 0.951498143405278233485002290026 ["ModularForm/GL2/Q/holomorphic/3675/1/k/b/2057/4"] "2-3675-105.62-c0-0-5" 1.3542761614332153 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1.74634880845100679009755544121 ["ModularForm/GL2/Q/holomorphic/3675/1/k/c/2057/4"] "2-3675-105.68-c0-0-0" 1.3542761614332153 1.834063921426284 2 3675 "105.68" [] [[0.0, 0.0]] 0 true true false false 0.44347609455230197 0 0.02036363190131954997820051477 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/a/68/1"] "2-3675-105.68-c0-0-1" 1.3542761614332153 1.834063921426284 2 3675 "105.68" [] [[0.0, 0.0]] 0 true true false false 0.4233787315225046 0 0.11929427102600731987036377169 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/d/68/3"] "2-3675-105.68-c0-0-10" 1.3542761614332153 1.834063921426284 2 3675 "105.68" [] [[0.0, 0.0]] 0 true true false false -0.48991343165138845 0 1.67128281796061836104076958495 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/d/68/1"] "2-3675-105.68-c0-0-11" 1.3542761614332153 1.834063921426284 2 3675 "105.68" [] [[0.0, 0.0]] 0 true true false false 0.3021463122774573 0 2.27096231468719585151716244956 ["ModularForm/GL2/Q/holomorphic/3675/1/bf/d/68/4"] "2-3675-105.68-c0-0-2" 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1.834063921426284 2 3675 "735.83" [] [[0.0, 0.0]] 0 true true false false 0.17465100645342463 0 1.02992279943480553107589807325 ["ModularForm/GL2/Q/holomorphic/3675/1/bx/a/818/1"] # 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).