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Properties

Label 1-712-712.501-r1-0-0
Degree $1$
Conductor $712$
Sign $0.155 - 0.987i$
Analytic cond. $76.5150$
Root an. cond. $76.5150$
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  + (0.977 − 0.212i)3-s + (0.755 + 0.654i)5-s + (−0.0713 − 0.997i)7-s + (0.909 − 0.415i)9-s + (−0.654 − 0.755i)11-s + (0.212 + 0.977i)13-s + (0.877 + 0.479i)15-s + (0.281 − 0.959i)17-s + (0.349 − 0.936i)19-s + (−0.281 − 0.959i)21-s + (−0.936 − 0.349i)23-s + (0.142 + 0.989i)25-s + (0.800 − 0.599i)27-s + (0.997 − 0.0713i)29-s + (−0.936 + 0.349i)31-s + ⋯
L(s)  = 1  + (0.977 − 0.212i)3-s + (0.755 + 0.654i)5-s + (−0.0713 − 0.997i)7-s + (0.909 − 0.415i)9-s + (−0.654 − 0.755i)11-s + (0.212 + 0.977i)13-s + (0.877 + 0.479i)15-s + (0.281 − 0.959i)17-s + (0.349 − 0.936i)19-s + (−0.281 − 0.959i)21-s + (−0.936 − 0.349i)23-s + (0.142 + 0.989i)25-s + (0.800 − 0.599i)27-s + (0.997 − 0.0713i)29-s + (−0.936 + 0.349i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.155 - 0.987i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ 0.155 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.493457691 - 2.132217450i\)
\(L(\frac12)\) \(\approx\) \(2.493457691 - 2.132217450i\)
\(L(1)\) \(\approx\) \(1.618566093 - 0.4044698602i\)
\(L(1)\) \(\approx\) \(1.618566093 - 0.4044698602i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.977 - 0.212i)T \)
5 \( 1 + (0.755 + 0.654i)T \)
7 \( 1 + (-0.0713 - 0.997i)T \)
11 \( 1 + (-0.654 - 0.755i)T \)
13 \( 1 + (0.212 + 0.977i)T \)
17 \( 1 + (0.281 - 0.959i)T \)
19 \( 1 + (0.349 - 0.936i)T \)
23 \( 1 + (-0.936 - 0.349i)T \)
29 \( 1 + (0.997 - 0.0713i)T \)
31 \( 1 + (-0.936 + 0.349i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (0.212 - 0.977i)T \)
43 \( 1 + (0.997 + 0.0713i)T \)
47 \( 1 + (0.540 + 0.841i)T \)
53 \( 1 + (0.540 - 0.841i)T \)
59 \( 1 + (-0.977 - 0.212i)T \)
61 \( 1 + (-0.599 - 0.800i)T \)
67 \( 1 + (-0.841 - 0.540i)T \)
71 \( 1 + (0.755 - 0.654i)T \)
73 \( 1 + (-0.415 + 0.909i)T \)
79 \( 1 + (-0.909 - 0.415i)T \)
83 \( 1 + (0.877 - 0.479i)T \)
97 \( 1 + (-0.654 + 0.755i)T \)
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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

−22.30637161449485424471535929890, −21.589460234885909427381375567758, −20.913380441549654707313614102421, −20.279436367812093905695539472, −19.50927735956682188221385345961, −18.34767166191802122228904578732, −17.980321720718999062286956207112, −16.7744414167726054532161859443, −15.79466118974942767208304181416, −15.24026441494521311135053855832, −14.37746761492572440928506721817, −13.45804279842943307524713096226, −12.66746280917768123833383007700, −12.18313445510477289098880859200, −10.38626386404396458728464559198, −10.00006009683986229342105725309, −9.03105922206520293585112756707, −8.2885969573345952583703276203, −7.61259495316042926887855446849, −6.038950753127725045174002911263, −5.40313214100470805348771170938, −4.333244087959622581093079844216, −3.13933137751380530048150306859, −2.21178770112183776897115379578, −1.405216788354931578858981461059, 0.61852537238569469228570458227, 1.88473658438263017873876304635, 2.816249899896536402657171248098, 3.63295048124480198476750105108, 4.758973404338953515723666363846, 6.0942077899757318005964300105, 7.04510801805668124463465539778, 7.56482457146018357383231185935, 8.80930112919305590825955334501, 9.51898306522302640646489712346, 10.42146420434252168168188557470, 11.1252157781408737109511671443, 12.44741439257696958724975277045, 13.56614573622292493012414537695, 13.9320386812996308770742173111, 14.35136032514984903806060126168, 15.73732399911257951343358728461, 16.2911956234839063824951462355, 17.53601749770660911967072908776, 18.25006648537516899665560887646, 18.97525433840544494411098582812, 19.75253185420848820715603482489, 20.672305597273699488938070611713, 21.26052076513807499412953069904, 22.03220172910523906930605329282

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