Properties

Label 1-23e2-529.340-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: $\chi_{529} (340, \cdot )$
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

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

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