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Properties

Label 1-837-837.533-r0-0-0
Degree $1$
Conductor $837$
Sign $0.185 - 0.982i$
Analytic cond. $3.88701$
Root an. cond. $3.88701$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.766 − 0.642i)5-s + (−0.939 − 0.342i)7-s + (0.5 + 0.866i)8-s + 10-s + (0.173 + 0.984i)11-s + (−0.173 + 0.984i)13-s + (0.939 − 0.342i)14-s + (−0.939 − 0.342i)16-s + (−0.5 − 0.866i)17-s + 19-s + (−0.766 + 0.642i)20-s + (−0.766 − 0.642i)22-s + (0.173 − 0.984i)23-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.766 − 0.642i)5-s + (−0.939 − 0.342i)7-s + (0.5 + 0.866i)8-s + 10-s + (0.173 + 0.984i)11-s + (−0.173 + 0.984i)13-s + (0.939 − 0.342i)14-s + (−0.939 − 0.342i)16-s + (−0.5 − 0.866i)17-s + 19-s + (−0.766 + 0.642i)20-s + (−0.766 − 0.642i)22-s + (0.173 − 0.984i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.185 - 0.982i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (533, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.185 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3263816546 - 0.2704809577i\)
\(L(\frac12)\) \(\approx\) \(0.3263816546 - 0.2704809577i\)
\(L(1)\) \(\approx\) \(0.5381161697 + 0.02760606438i\)
\(L(1)\) \(\approx\) \(0.5381161697 + 0.02760606438i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 + (0.173 + 0.984i)T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.173 + 0.984i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 - T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (-0.939 - 0.342i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.939 - 0.342i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.04498665361822831098038330102, −21.75789255343161775402456441292, −20.43301175934748808553144546597, −19.6026239655516235148158593653, −19.3389367068258823549430293749, −18.46261858373347149642004737513, −17.75293763151564157874672056039, −16.77827786541043591284583953075, −15.77937274277416789216217145978, −15.54228564704274354799106663054, −14.15037605445508215699683162720, −13.12334456077608755439944817816, −12.39135251487705081361187191832, −11.56038419081482622222075001579, −10.80698802257021164160017080447, −10.1097243269280266083374131565, −9.120795113326545241199289989697, −8.321730007502615139936524517443, −7.4819455143746016897401110732, −6.630846848437795861588486327000, −5.58590609205817473311278231112, −3.92335058275129613719353267851, −3.26625383607427975524972204446, −2.63556252765234433883518525299, −1.02506985376940056491016141206, 0.30818671885763798119766642, 1.55279905901702452825750704910, 2.93353478266660917674580328527, 4.4179677994472481449922113935, 4.88466407594756474274389607001, 6.38609995404664112203039541900, 6.96764544711154059064663706232, 7.74519070585488291232459572759, 8.751458124597609622786589540468, 9.54190654048683741515243738420, 10.04684692764985240254901386273, 11.39833828954429007473220811391, 11.992316968707876141937311521639, 13.11464886705539117878110328694, 13.98224296155934482306910901448, 15.01039283346232896743760410895, 15.73970969365661716341915065608, 16.463552587815813132683337298234, 16.855750828648772612421406418206, 17.99914473445502614761214434856, 18.719200236443334423065820389029, 19.663466070987840796465361670928, 20.0366514915069897289958856608, 20.791760512120722271267822899687, 22.28665288880517630742842285793

Graph of the $Z$-function along the critical line