Properties

Label 1-712-712.283-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: $\chi_{712} (283, \cdot )$
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

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

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