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Properties

Label 1-712-712.365-r0-0-0
Degree $1$
Conductor $712$
Sign $0.746 + 0.665i$
Analytic cond. $3.30651$
Root an. cond. $3.30651$
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  + (−0.989 − 0.142i)3-s + (0.841 + 0.540i)5-s + (0.540 − 0.841i)7-s + (0.959 + 0.281i)9-s + (−0.841 + 0.540i)11-s + (0.989 + 0.142i)13-s + (−0.755 − 0.654i)15-s + (0.654 + 0.755i)17-s + (−0.281 + 0.959i)19-s + (−0.654 + 0.755i)21-s + (−0.281 + 0.959i)23-s + (0.415 + 0.909i)25-s + (−0.909 − 0.415i)27-s + (0.540 − 0.841i)29-s + (−0.281 − 0.959i)31-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)3-s + (0.841 + 0.540i)5-s + (0.540 − 0.841i)7-s + (0.959 + 0.281i)9-s + (−0.841 + 0.540i)11-s + (0.989 + 0.142i)13-s + (−0.755 − 0.654i)15-s + (0.654 + 0.755i)17-s + (−0.281 + 0.959i)19-s + (−0.654 + 0.755i)21-s + (−0.281 + 0.959i)23-s + (0.415 + 0.909i)25-s + (−0.909 − 0.415i)27-s + (0.540 − 0.841i)29-s + (−0.281 − 0.959i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.746 + 0.665i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ 0.746 + 0.665i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.176185537 + 0.4479766933i\)
\(L(\frac12)\) \(\approx\) \(1.176185537 + 0.4479766933i\)
\(L(1)\) \(\approx\) \(0.9849805620 + 0.1177015575i\)
\(L(1)\) \(\approx\) \(0.9849805620 + 0.1177015575i\)

Euler product

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

−22.24134618733647587530549321517, −21.7677947265241614752137149355, −20.96735737526109737004663065576, −20.50603522069016662595614568403, −18.877642463682779074271368527114, −18.18833668243613958193999867640, −17.811660913705905739763944231403, −16.719884534727075586211566188716, −16.08082572774131708216197734992, −15.38617535073752135608091936295, −14.142047934480549837029354104213, −13.27760976735053704545985455379, −12.49521712890984809606892327502, −11.69562659718497357808219732112, −10.7641822661137056459893982739, −10.12202422006845425854700420234, −8.91968388207279374122733824470, −8.35897121457946410613241123989, −6.88921052403986662514722146666, −5.99066512094119599164128099256, −5.22955743460798411540702411384, −4.77667698107968336416903269826, −3.13832944094176363859805426455, −1.91770568656644889880128374500, −0.781232287475212184601016321487, 1.24511393444816665739193648680, 2.01903913747217681355873884085, 3.62984600018320254341875187568, 4.58836475427737259823256274686, 5.76336545369772992914572872949, 6.15780935317359558223997908558, 7.39011726474361198228343219772, 7.95802498631383931554307571204, 9.58610905652247128598649712414, 10.39391919314063683477113723131, 10.79202096666377501289738531382, 11.754772233478781461896721997933, 12.86981133584876619458236094479, 13.48676445873240820868635881538, 14.364720608141287258745426435879, 15.36123509320049876366444702501, 16.34226896781794221716901046631, 17.18363346522578565787452650334, 17.7149768477721610162967060481, 18.427351294301057200175278578760, 19.15830282338289607618091866109, 20.62773374632363582404939226480, 21.12057403369847397415745413182, 21.81575240979754257570124698972, 23.03647501038678609672536967088

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