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Properties

Label 1-712-712.59-r0-0-0
Degree $1$
Conductor $712$
Sign $0.921 - 0.387i$
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  + (−0.997 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.921 - 0.387i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ 0.921 - 0.387i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.166328891 - 0.2353654791i\)
\(L(\frac12)\) \(\approx\) \(1.166328891 - 0.2353654791i\)
\(L(1)\) \(\approx\) \(0.9289856051 - 0.09963762176i\)
\(L(1)\) \(\approx\) \(0.9289856051 - 0.09963762176i\)

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

−22.49567190709856695575399780797, −22.05507041170249126683990285852, −21.26947051054003404049268809930, −20.23451247678452680311199290978, −19.09266116018914985236497689536, −18.62025296915251107209678404632, −17.67160446770411298605825267439, −17.03547066730856567077523885153, −16.276045215012302447332402217473, −15.2649081664336338745495191574, −14.26030275111353067437579771037, −13.87904221289190463359469336976, −12.3495751189719539604592051203, −11.66450584730205029285278658670, −11.10826476331263011927149816677, −10.34036605941782520590771081719, −9.35862226630670373963114257831, −7.9844451461469354302655939678, −7.15441076588981300446616241539, −6.49236728962313329095782807270, −5.52179047187451610541740657162, −4.35337855702091559838484403464, −3.706821851466269292805691295811, −2.09956182985320344864024670720, −0.99219494542591751676686389315, 0.93520392718410982327815052941, 1.71165702299974023400724171949, 3.55584157248532896654820548443, 4.51525406010637079441446039525, 5.442417329185829591406503910410, 5.81937031619852114748772852806, 7.40871686863558522738675286232, 7.950941953485663252668310195359, 9.22228675142973441509237652049, 9.87391364536140412343098610534, 11.12537516511638157252217456450, 11.8270906667322387898210911987, 12.31155614118685862435747354145, 13.19654520023467908094071758873, 14.47854066895747869938248239069, 15.22982225060689105971226641098, 16.28425191836342418460128116167, 16.69409143535873202132403938017, 17.78926062415698213547587133692, 18.08271112421989499163120615658, 19.3098132607104436855351608553, 20.32036010197668878653232426123, 20.87890651307320753307772965237, 21.85984851034159223871641381600, 22.510177942817755405626168821875

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