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Properties

Label 1-23e2-529.133-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: $\chi_{529} (133, \cdot )$
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

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

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