Properties

Label 1-23e2-529.515-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.967 + 0.251i$
Analytic cond. $2.45666$
Root an. cond. $2.45666$
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  + (0.969 + 0.245i)2-s + (−0.820 + 0.571i)3-s + (0.879 + 0.476i)4-s + (0.586 + 0.809i)5-s + (−0.935 + 0.352i)6-s + (−0.709 + 0.704i)7-s + (0.735 + 0.678i)8-s + (0.346 − 0.938i)9-s + (0.369 + 0.929i)10-s + (−0.926 + 0.375i)11-s + (−0.993 + 0.111i)12-s + (−0.972 − 0.233i)13-s + (−0.860 + 0.508i)14-s + (−0.944 − 0.329i)15-s + (0.545 + 0.837i)16-s + (−0.616 + 0.787i)17-s + ⋯
L(s)  = 1  + (0.969 + 0.245i)2-s + (−0.820 + 0.571i)3-s + (0.879 + 0.476i)4-s + (0.586 + 0.809i)5-s + (−0.935 + 0.352i)6-s + (−0.709 + 0.704i)7-s + (0.735 + 0.678i)8-s + (0.346 − 0.938i)9-s + (0.369 + 0.929i)10-s + (−0.926 + 0.375i)11-s + (−0.993 + 0.111i)12-s + (−0.972 − 0.233i)13-s + (−0.860 + 0.508i)14-s + (−0.944 − 0.329i)15-s + (0.545 + 0.837i)16-s + (−0.616 + 0.787i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.967 + 0.251i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.967 + 0.251i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1891920322 + 1.479998840i\)
\(L(\frac12)\) \(\approx\) \(0.1891920322 + 1.479998840i\)
\(L(1)\) \(\approx\) \(0.9994932140 + 0.8495170113i\)
\(L(1)\) \(\approx\) \(0.9994932140 + 0.8495170113i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.969 + 0.245i)T \)
3 \( 1 + (-0.820 + 0.571i)T \)
5 \( 1 + (0.586 + 0.809i)T \)
7 \( 1 + (-0.709 + 0.704i)T \)
11 \( 1 + (-0.926 + 0.375i)T \)
13 \( 1 + (-0.972 - 0.233i)T \)
17 \( 1 + (-0.616 + 0.787i)T \)
19 \( 1 + (0.948 - 0.317i)T \)
29 \( 1 + (-0.982 - 0.185i)T \)
31 \( 1 + (-0.404 - 0.914i)T \)
37 \( 1 + (0.988 + 0.148i)T \)
41 \( 1 + (-0.966 - 0.257i)T \)
43 \( 1 + (0.940 + 0.340i)T \)
47 \( 1 + (0.962 + 0.269i)T \)
53 \( 1 + (0.369 - 0.929i)T \)
59 \( 1 + (0.105 + 0.994i)T \)
61 \( 1 + (-0.885 + 0.465i)T \)
67 \( 1 + (-0.0434 + 0.999i)T \)
71 \( 1 + (0.437 - 0.899i)T \)
73 \( 1 + (-0.982 + 0.185i)T \)
79 \( 1 + (0.606 + 0.794i)T \)
83 \( 1 + (0.0558 + 0.998i)T \)
89 \( 1 + (-0.263 - 0.964i)T \)
97 \( 1 + (-0.673 + 0.739i)T \)
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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

−23.21534631031744874862620315908, −22.164990084790811207592421501553, −21.78636987914413439411669995035, −20.53028530767574195429714811605, −20.03052471760119530165368456942, −18.976960512385432025833766806495, −18.036237892341786917773020674055, −16.87122718060278102954157416858, −16.394380199603840913500909040799, −15.631491732661104255153077219390, −14.02164275882693633947463159608, −13.54341797783449258926510701523, −12.77278205865009366571497201095, −12.18856248467910160939513702702, −11.12601285788921246105201386726, −10.271908690821389709129421673249, −9.41959145851158211799020354430, −7.63709427016142602821702882835, −6.98420196534716622694091638144, −5.88527195339888728914521097425, −5.21454887586757877030427335765, −4.42275017039942734713865306547, −2.93661557621558800379522718375, −1.84643276547300877197856315462, −0.584364242957796280140992608498, 2.213238901464258884399578862365, 2.990676192764682579757125891492, 4.15148241126184691630046929910, 5.37475518205536853429784723281, 5.778121225674434021664479944367, 6.75875898810481797937713573673, 7.58440452898700961803622459616, 9.34102946983507083791431359361, 10.15480870589877597325736769401, 10.97344713420681650796239185944, 11.87510941022703032395484388666, 12.781806951404058201502263980319, 13.43419326805795912874390347233, 14.92385778025507604059435223212, 15.12372536287484702242178868949, 16.03368371221041591427829612336, 16.97454862683267757133747453525, 17.78283975098952522026142733537, 18.65211111588757396278388633504, 19.91111431180454358685050395102, 20.93017235912578858898262477550, 21.719294310415804368529812673324, 22.40684857623083815909680146566, 22.541311916080702558711068698705, 23.75973738033706165214808312829

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