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Properties

Label 1-712-712.181-r1-0-0
Degree $1$
Conductor $712$
Sign $-0.375 - 0.927i$
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  + (−0.997 − 0.0713i)3-s + (−0.281 + 0.959i)5-s + (0.877 − 0.479i)7-s + (0.989 + 0.142i)9-s + (−0.959 + 0.281i)11-s + (0.0713 − 0.997i)13-s + (0.349 − 0.936i)15-s + (0.909 + 0.415i)17-s + (0.800 − 0.599i)19-s + (−0.909 + 0.415i)21-s + (−0.599 − 0.800i)23-s + (−0.841 − 0.540i)25-s + (−0.977 − 0.212i)27-s + (0.479 + 0.877i)29-s + (−0.599 + 0.800i)31-s + ⋯
L(s)  = 1  + (−0.997 − 0.0713i)3-s + (−0.281 + 0.959i)5-s + (0.877 − 0.479i)7-s + (0.989 + 0.142i)9-s + (−0.959 + 0.281i)11-s + (0.0713 − 0.997i)13-s + (0.349 − 0.936i)15-s + (0.909 + 0.415i)17-s + (0.800 − 0.599i)19-s + (−0.909 + 0.415i)21-s + (−0.599 − 0.800i)23-s + (−0.841 − 0.540i)25-s + (−0.977 − 0.212i)27-s + (0.479 + 0.877i)29-s + (−0.599 + 0.800i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.375 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.375 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.375 - 0.927i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ -0.375 - 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3613032756 - 0.5359196963i\)
\(L(\frac12)\) \(\approx\) \(0.3613032756 - 0.5359196963i\)
\(L(1)\) \(\approx\) \(0.7373525971 + 0.009595421311i\)
\(L(1)\) \(\approx\) \(0.7373525971 + 0.009595421311i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (-0.997 - 0.0713i)T \)
5 \( 1 + (-0.281 + 0.959i)T \)
7 \( 1 + (0.877 - 0.479i)T \)
11 \( 1 + (-0.959 + 0.281i)T \)
13 \( 1 + (0.0713 - 0.997i)T \)
17 \( 1 + (0.909 + 0.415i)T \)
19 \( 1 + (0.800 - 0.599i)T \)
23 \( 1 + (-0.599 - 0.800i)T \)
29 \( 1 + (0.479 + 0.877i)T \)
31 \( 1 + (-0.599 + 0.800i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (0.0713 + 0.997i)T \)
43 \( 1 + (0.479 - 0.877i)T \)
47 \( 1 + (-0.755 - 0.654i)T \)
53 \( 1 + (-0.755 + 0.654i)T \)
59 \( 1 + (0.997 - 0.0713i)T \)
61 \( 1 + (-0.212 + 0.977i)T \)
67 \( 1 + (0.654 + 0.755i)T \)
71 \( 1 + (-0.281 - 0.959i)T \)
73 \( 1 + (0.142 + 0.989i)T \)
79 \( 1 + (-0.989 + 0.142i)T \)
83 \( 1 + (0.349 + 0.936i)T \)
97 \( 1 + (-0.959 - 0.281i)T \)
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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

−22.759290662630388397025051082370, −21.69473945567102774838402605981, −20.998027206232522910017479863055, −20.64457715490098521117766740886, −19.144056745047720501539982087069, −18.5362666322594001779216859855, −17.676308571703499763908688689637, −16.9265470595165326808981718723, −16.01211645162624019116789231201, −15.72847112283837554169371884950, −14.381239579254444945922274127596, −13.4451762218222756529710852426, −12.447924729226765045996834380798, −11.734254636562652309444490332277, −11.3206590441609735961162279187, −10.04322710636438029936990850500, −9.29345419429892849955057202569, −8.05121060427601881644923861028, −7.55740857851415969290665500930, −6.05933224581592438738066755853, −5.32223120426253594461005020374, −4.74798404856100919786459760222, −3.678527900158619260733616358711, −1.95642408402326030349658315080, −1.04287341186425901962231849798, 0.2008319196492718044433714076, 1.39311163574692589055658859476, 2.74961851620224864093981616362, 3.87343313624097456579026252121, 5.03921103290132407784728748212, 5.63159882512722986005074266969, 6.88968834745770435918313447101, 7.50728660098808329692860396743, 8.27446815615963825690193484204, 10.06464661957696332922299698434, 10.50853033973090259992377041563, 11.11782112585500067620240964826, 12.08012155986751875409416232640, 12.84972598551560786697813715944, 13.96772439796830298898640552563, 14.78236636719543368808663288136, 15.658694004098563906361585701997, 16.36771205441676962337998736498, 17.552572458268870180526188640894, 18.018636896166357532577428524665, 18.51605339773043978884646510239, 19.695489034533954245493533189296, 20.615437301680866554229302072704, 21.49775876810958318340180390625, 22.23732320486395339021326925873

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