Properties

Label 1-712-712.27-r0-0-0
Degree $1$
Conductor $712$
Sign $0.838 + 0.545i$
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  + (0.977 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.838 + 0.545i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ 0.838 + 0.545i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.304638441 + 0.6835253466i\)
\(L(\frac12)\) \(\approx\) \(2.304638441 + 0.6835253466i\)
\(L(1)\) \(\approx\) \(1.667226038 + 0.2360826619i\)
\(L(1)\) \(\approx\) \(1.667226038 + 0.2360826619i\)

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

−22.435455646676803731702388939423, −21.78381102089764490789283175806, −20.5214209969338700521990927412, −20.24565534676266466660784567851, −19.47578912944786298873238121228, −18.27537173806659520459851232799, −17.89338996510944919459958730905, −16.924321344042177883322948130718, −15.85022255508102452960438830012, −14.71608620744168774992025436367, −14.441693263106964034559936493294, −13.411576088252598394145007092868, −12.99409373253334373496696693374, −11.70631483924311716138042050526, −10.55493706437514523689035926444, −9.791405378026714089595908549183, −9.297048157776788820105587420136, −7.91904538109268161300843459365, −7.19629962712657294744127333649, −6.6086772665693183020052633270, −5.22188330466842924810216704286, −4.04812505159951379393740820443, −3.12432028754599740889137492094, −2.252651802447792044102159695039, −1.11861354653392158750997845230, 1.62024421335172446213278337199, 2.11522111517696035272837867619, 3.47643034175716485574289036067, 4.27495883977985355970023303511, 5.61167066513149950042932839128, 6.15899652652038803469632199492, 7.627871878238406595263177132336, 8.51124541615026435183879541100, 9.27021345489501639326517577628, 9.6219325827702359198349695600, 10.8927912196175572932160289353, 12.13024652453782199751262123089, 12.72451224424624679579868775012, 13.81653949434207493018326857309, 14.31397952461714394731159953690, 15.14231611494850756152344509693, 16.28637822692241166577749704432, 16.655791824340757160811701333920, 17.90315623633814772763164775526, 18.839969784002018575079738316683, 19.34308840153350321180511795020, 20.35017086728867887797850618446, 21.109605852560259104045772658, 21.7779441250758046492822610918, 22.17377370869442129248967618523

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