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Properties

Label 1-712-712.363-r0-0-0
Degree $1$
Conductor $712$
Sign $0.769 - 0.638i$
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  + (0.877 − 0.479i)3-s + (0.909 − 0.415i)5-s + (0.936 + 0.349i)7-s + (0.540 − 0.841i)9-s + (−0.415 + 0.909i)11-s + (−0.479 − 0.877i)13-s + (0.599 − 0.800i)15-s + (−0.989 − 0.142i)17-s + (0.212 + 0.977i)19-s + (0.989 − 0.142i)21-s + (0.977 − 0.212i)23-s + (0.654 − 0.755i)25-s + (0.0713 − 0.997i)27-s + (0.349 − 0.936i)29-s + (0.977 + 0.212i)31-s + ⋯
L(s)  = 1  + (0.877 − 0.479i)3-s + (0.909 − 0.415i)5-s + (0.936 + 0.349i)7-s + (0.540 − 0.841i)9-s + (−0.415 + 0.909i)11-s + (−0.479 − 0.877i)13-s + (0.599 − 0.800i)15-s + (−0.989 − 0.142i)17-s + (0.212 + 0.977i)19-s + (0.989 − 0.142i)21-s + (0.977 − 0.212i)23-s + (0.654 − 0.755i)25-s + (0.0713 − 0.997i)27-s + (0.349 − 0.936i)29-s + (0.977 + 0.212i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.769 - 0.638i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ 0.769 - 0.638i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.341968287 - 0.8455907147i\)
\(L(\frac12)\) \(\approx\) \(2.341968287 - 0.8455907147i\)
\(L(1)\) \(\approx\) \(1.689486244 - 0.3618790177i\)
\(L(1)\) \(\approx\) \(1.689486244 - 0.3618790177i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.877 - 0.479i)T \)
5 \( 1 + (0.909 - 0.415i)T \)
7 \( 1 + (0.936 + 0.349i)T \)
11 \( 1 + (-0.415 + 0.909i)T \)
13 \( 1 + (-0.479 - 0.877i)T \)
17 \( 1 + (-0.989 - 0.142i)T \)
19 \( 1 + (0.212 + 0.977i)T \)
23 \( 1 + (0.977 - 0.212i)T \)
29 \( 1 + (0.349 - 0.936i)T \)
31 \( 1 + (0.977 + 0.212i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (-0.479 + 0.877i)T \)
43 \( 1 + (-0.349 - 0.936i)T \)
47 \( 1 + (0.281 + 0.959i)T \)
53 \( 1 + (-0.281 + 0.959i)T \)
59 \( 1 + (-0.877 - 0.479i)T \)
61 \( 1 + (0.997 + 0.0713i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (-0.909 - 0.415i)T \)
73 \( 1 + (-0.841 + 0.540i)T \)
79 \( 1 + (0.540 + 0.841i)T \)
83 \( 1 + (0.599 + 0.800i)T \)
97 \( 1 + (0.415 + 0.909i)T \)
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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

−22.30230167758130796836992248441, −21.67087790128475811433431800849, −21.17475268582009386713204407580, −20.43294513491242940634337785745, −19.43472412768140344205116753964, −18.74224931379712694952734836114, −17.75081030250512214623501278800, −17.053767545711369972700779631058, −16.06305574667592949792682377652, −15.07757951576757515449643757361, −14.45243056553982056668109430579, −13.534361845303537177577622416044, −13.37393706762981580809817269358, −11.61476640437378844003212076443, −10.81575490106711216121613959527, −10.17166140388136971499634006565, −9.01929332267601445900709292342, −8.61851323164237683490589535167, −7.36837318118885020463133289649, −6.61605281723544452500043928270, −5.15666004329881333227661312213, −4.61378437256639784804025323554, −3.26844180283411839876068759448, −2.431111018413061343563136708797, −1.4951010087371902700683682218, 1.23883730633060583018852947888, 2.1466944681192920373313253551, 2.81347318092885706519411314203, 4.41262648856083565521807595143, 5.162550056644654137000679126216, 6.26727413600889710575724373901, 7.36875042719512683424834144334, 8.151411234692184035220791429639, 8.91714493151773260053391507504, 9.799677223974069282242308380677, 10.58186146066220212518528571607, 12.0198227573492773001711547535, 12.64799525822613294903584981314, 13.457324085896184880437089901329, 14.20978991748933478237347150887, 15.042768109940205491165351982546, 15.62712247263889114371544923184, 17.16764861041815792212783714579, 17.70663244311408255106537136255, 18.29838591892591203909400896833, 19.2775608919047873961772854796, 20.42485869522738078812917142052, 20.647044563844697545523545639805, 21.42643645317751585841013519016, 22.4673038943953203533910968434

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