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Properties

Label 1-920-920.677-r0-0-0
Degree $1$
Conductor $920$
Sign $0.912 - 0.408i$
Analytic cond. $4.27246$
Root an. cond. $4.27246$
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  + (−0.909 + 0.415i)3-s + (0.540 − 0.841i)7-s + (0.654 − 0.755i)9-s + (−0.142 − 0.989i)11-s + (0.540 + 0.841i)13-s + (0.281 + 0.959i)17-s + (0.959 + 0.281i)19-s + (−0.142 + 0.989i)21-s + (−0.281 + 0.959i)27-s + (−0.959 + 0.281i)29-s + (0.415 − 0.909i)31-s + (0.540 + 0.841i)33-s + (−0.755 − 0.654i)37-s + (−0.841 − 0.540i)39-s + (−0.654 − 0.755i)41-s + ⋯
L(s)  = 1  + (−0.909 + 0.415i)3-s + (0.540 − 0.841i)7-s + (0.654 − 0.755i)9-s + (−0.142 − 0.989i)11-s + (0.540 + 0.841i)13-s + (0.281 + 0.959i)17-s + (0.959 + 0.281i)19-s + (−0.142 + 0.989i)21-s + (−0.281 + 0.959i)27-s + (−0.959 + 0.281i)29-s + (0.415 − 0.909i)31-s + (0.540 + 0.841i)33-s + (−0.755 − 0.654i)37-s + (−0.841 − 0.540i)39-s + (−0.654 − 0.755i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.912 - 0.408i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (677, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ 0.912 - 0.408i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.143549816 - 0.2443507903i\)
\(L(\frac12)\) \(\approx\) \(1.143549816 - 0.2443507903i\)
\(L(1)\) \(\approx\) \(0.9137035117 - 0.03259288635i\)
\(L(1)\) \(\approx\) \(0.9137035117 - 0.03259288635i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.909 + 0.415i)T \)
7 \( 1 + (0.540 - 0.841i)T \)
11 \( 1 + (-0.142 - 0.989i)T \)
13 \( 1 + (0.540 + 0.841i)T \)
17 \( 1 + (0.281 + 0.959i)T \)
19 \( 1 + (0.959 + 0.281i)T \)
29 \( 1 + (-0.959 + 0.281i)T \)
31 \( 1 + (0.415 - 0.909i)T \)
37 \( 1 + (-0.755 - 0.654i)T \)
41 \( 1 + (-0.654 - 0.755i)T \)
43 \( 1 + (0.909 - 0.415i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.540 - 0.841i)T \)
59 \( 1 + (0.841 - 0.540i)T \)
61 \( 1 + (0.415 - 0.909i)T \)
67 \( 1 + (-0.989 - 0.142i)T \)
71 \( 1 + (-0.142 + 0.989i)T \)
73 \( 1 + (-0.281 + 0.959i)T \)
79 \( 1 + (0.841 - 0.540i)T \)
83 \( 1 + (0.755 + 0.654i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (-0.755 + 0.654i)T \)
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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

−22.25970712971443466269261783629, −21.10255237464070614403485704124, −20.5181799889083410211942032596, −19.471435813654485006743232396496, −18.325132383596953915722190932062, −18.14443218182620940919215877940, −17.41982943827310430117714174841, −16.37223823332500751752541075178, −15.60966005576044018290927967883, −14.954996895037066805955101654064, −13.74901752580998001208184089769, −13.01037075453080497461166999545, −12.00602209721055691164242370573, −11.73225061533792460583386900428, −10.65951841847245303000657290602, −9.870126913787832330244158635994, −8.82168689428293631929905982518, −7.73027361373566825802549949549, −7.14735108203469922825362255743, −5.997929641626199253939776690729, −5.25132784242300241588518322366, −4.6704416367603045663295704560, −3.130840336472634122596032218443, −2.03600387251441490939349510656, −1.01685548156210694709365246466, 0.7540964170329672408318724525, 1.755174999935870335744969865532, 3.59628382024407171688062227542, 4.01516560920486824812369674070, 5.21156137003763860110030026194, 5.89803072091552262123859708192, 6.840152889350214243538108656539, 7.76653869440218662972611365730, 8.78647331532088313579306423129, 9.78461333310932956160376684593, 10.68120993271486482290167130257, 11.19758312971703081132899790075, 11.913196527852927672616064876306, 12.997211628751706900750169746040, 13.8639478397018207077211866042, 14.5963614276754183187462190290, 15.713894992940596309563931807074, 16.38891294564996307385937707883, 17.00054694072163239031783596308, 17.70545937549167960160532628190, 18.64566954334812004196058835348, 19.29257440670225769104937967642, 20.74799303507720965566544521227, 20.85827714970371956138247895086, 21.915478485514667554993640357930

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