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Properties

Label 1-23e2-529.354-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.173 - 0.984i$
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  + (0.179 − 0.983i)2-s + (0.626 + 0.779i)3-s + (−0.935 − 0.352i)4-s + (0.0558 − 0.998i)5-s + (0.879 − 0.476i)6-s + (0.798 − 0.601i)7-s + (−0.514 + 0.857i)8-s + (−0.215 + 0.976i)9-s + (−0.972 − 0.233i)10-s + (−0.0434 − 0.999i)11-s + (−0.311 − 0.950i)12-s + (0.369 + 0.929i)13-s + (−0.449 − 0.893i)14-s + (0.813 − 0.581i)15-s + (0.751 + 0.659i)16-s + (0.998 + 0.0496i)17-s + ⋯
L(s)  = 1  + (0.179 − 0.983i)2-s + (0.626 + 0.779i)3-s + (−0.935 − 0.352i)4-s + (0.0558 − 0.998i)5-s + (0.879 − 0.476i)6-s + (0.798 − 0.601i)7-s + (−0.514 + 0.857i)8-s + (−0.215 + 0.976i)9-s + (−0.972 − 0.233i)10-s + (−0.0434 − 0.999i)11-s + (−0.311 − 0.950i)12-s + (0.369 + 0.929i)13-s + (−0.449 − 0.893i)14-s + (0.813 − 0.581i)15-s + (0.751 + 0.659i)16-s + (0.998 + 0.0496i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (354, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.160745713 - 1.382733188i\)
\(L(\frac12)\) \(\approx\) \(1.160745713 - 1.382733188i\)
\(L(1)\) \(\approx\) \(1.194891872 - 0.7156134056i\)
\(L(1)\) \(\approx\) \(1.194891872 - 0.7156134056i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.179 - 0.983i)T \)
3 \( 1 + (0.626 + 0.779i)T \)
5 \( 1 + (0.0558 - 0.998i)T \)
7 \( 1 + (0.798 - 0.601i)T \)
11 \( 1 + (-0.0434 - 0.999i)T \)
13 \( 1 + (0.369 + 0.929i)T \)
17 \( 1 + (0.998 + 0.0496i)T \)
19 \( 1 + (-0.381 - 0.924i)T \)
29 \( 1 + (0.503 - 0.863i)T \)
31 \( 1 + (0.130 - 0.991i)T \)
37 \( 1 + (0.912 + 0.409i)T \)
41 \( 1 + (-0.673 + 0.739i)T \)
43 \( 1 + (-0.998 + 0.0620i)T \)
47 \( 1 + (0.962 - 0.269i)T \)
53 \( 1 + (-0.972 + 0.233i)T \)
59 \( 1 + (-0.999 + 0.0372i)T \)
61 \( 1 + (0.980 + 0.197i)T \)
67 \( 1 + (-0.926 - 0.375i)T \)
71 \( 1 + (0.999 - 0.0248i)T \)
73 \( 1 + (0.503 + 0.863i)T \)
79 \( 1 + (-0.873 - 0.487i)T \)
83 \( 1 + (0.586 - 0.809i)T \)
89 \( 1 + (0.524 - 0.851i)T \)
97 \( 1 + (-0.966 - 0.257i)T \)
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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

−23.49750973020456526417708460870, −23.31710030962623603171591178253, −22.26897899972422555274032958517, −21.29594955641196985054891999799, −20.42176917570211554745849571066, −19.16324130667829689467655439979, −18.32734634079687051434030139000, −18.023206137739668426450760417891, −17.19057748936303250807692860054, −15.72644166663278964647810089927, −14.92498676536958054687825647664, −14.53033851512444522047712942768, −13.78785035193337225078015113744, −12.57929958475035398344891186481, −12.10134790849559554889483332294, −10.5541928203478961646212391599, −9.54836529866928743258932846246, −8.36215948297114884365421258488, −7.82098653845921154715332990188, −7.01687583223731071703363994419, −6.093510892866283992007175728892, −5.18706214403054444685838643231, −3.731106988969658698438616391, −2.81117441518356002729442443466, −1.52361392237911579215470698160, 0.944167412979144149718004298616, 2.05572211557567432337312890214, 3.31937239495822893752830739748, 4.30131166097094908263008191913, 4.78849675546946515231770945543, 5.89991183855751133011060783218, 7.94771340885087554149364628908, 8.52095440373094181140817863255, 9.35733237696600203415343569626, 10.17394207898825077054799156183, 11.22317738757345949201684033692, 11.73077036029535001183362431235, 13.202905674548731822007225083944, 13.69048972908537838330512178538, 14.41797104680529985805027637185, 15.47802830571623604782884838571, 16.708103228214864240822995923443, 17.081636669543362898996418953398, 18.522739038463911769211737596031, 19.3328973277896962352095886079, 20.13722712741395053346719480372, 20.79801285962475556016965052473, 21.39892963908695883564158341676, 21.85899902413293558794997169482, 23.35569476868489892454246308582

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