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Properties

Label 1-712-712.395-r1-0-0
Degree $1$
Conductor $712$
Sign $-0.896 + 0.443i$
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  + (−0.142 + 0.989i)3-s + (−0.841 − 0.540i)5-s + (−0.841 − 0.540i)7-s + (−0.959 − 0.281i)9-s + (0.841 − 0.540i)11-s + (0.142 − 0.989i)13-s + (0.654 − 0.755i)15-s + (−0.654 − 0.755i)17-s + (−0.959 − 0.281i)19-s + (0.654 − 0.755i)21-s + (0.959 + 0.281i)23-s + (0.415 + 0.909i)25-s + (0.415 − 0.909i)27-s + (−0.841 − 0.540i)29-s + (0.959 − 0.281i)31-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)3-s + (−0.841 − 0.540i)5-s + (−0.841 − 0.540i)7-s + (−0.959 − 0.281i)9-s + (0.841 − 0.540i)11-s + (0.142 − 0.989i)13-s + (0.654 − 0.755i)15-s + (−0.654 − 0.755i)17-s + (−0.959 − 0.281i)19-s + (0.654 − 0.755i)21-s + (0.959 + 0.281i)23-s + (0.415 + 0.909i)25-s + (0.415 − 0.909i)27-s + (−0.841 − 0.540i)29-s + (0.959 − 0.281i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.896 + 0.443i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ -0.896 + 0.443i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02743137032 - 0.1173592462i\)
\(L(\frac12)\) \(\approx\) \(0.02743137032 - 0.1173592462i\)
\(L(1)\) \(\approx\) \(0.6749661780 - 0.04001585562i\)
\(L(1)\) \(\approx\) \(0.6749661780 - 0.04001585562i\)

Euler product

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

−22.89512903497118842667622776949, −22.350554486702785205189543187525, −21.38408212499191991958756135462, −20.029395604243943235792470881418, −19.36083297038542381700460543605, −18.99380306570253641699171489593, −18.20151588361718526540948313170, −17.11822992333003007076760779045, −16.490714771890801136337666960494, −15.27943120436841870575686154762, −14.739064998049416547101385417877, −13.71707903893377874922565704876, −12.69447967909889692381530573616, −12.1823614726869578533566466024, −11.37731885881101566663983931117, −10.52818844187707111724615240308, −9.10832148324656874106220746874, −8.54455819680074503723245907861, −7.326691296207517214665048922429, −6.6138165350171517225196032084, −6.18389651918930347328308795099, −4.58785746114542521882205750832, −3.574493393871361374923242584785, −2.51560756326404431458812124913, −1.50303526686189401945398344889, 0.04002402883633571086289489668, 0.77520284760991437500208923120, 2.85450542793223249116174384092, 3.711683142302137148299574589876, 4.35089305203985989326382824588, 5.38966551032523732068312976629, 6.42518206026278702712170971980, 7.460816531874815718205081947476, 8.70265536706206846207534628108, 9.131777954192478197923342637978, 10.23736146116251869256336570096, 11.03965526736430258718852346560, 11.74733039417878286285680833730, 12.81203127279088326221040154728, 13.60785523293029569212221576746, 14.75357893965709582812831388434, 15.640935556730022575853599872302, 15.98264646588078647068643369917, 17.07278548197813104292786458640, 17.339778981227635228734980995660, 19.12631798368697734586250292031, 19.47133198915724520329098340698, 20.49286787966130163005010203470, 20.84512006157285054244168761105, 22.219853199330818337804737461450

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