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Properties

Label 1-23e2-529.154-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.826 - 0.562i$
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  + (−0.998 − 0.0620i)2-s + (−0.806 − 0.591i)3-s + (0.992 + 0.123i)4-s + (−0.972 − 0.233i)5-s + (0.767 + 0.640i)6-s + (−0.556 + 0.831i)7-s + (−0.982 − 0.185i)8-s + (0.299 + 0.954i)9-s + (0.955 + 0.293i)10-s + (−0.635 + 0.771i)11-s + (−0.726 − 0.687i)12-s + (0.664 + 0.747i)13-s + (0.606 − 0.794i)14-s + (0.645 + 0.763i)15-s + (0.969 + 0.245i)16-s + (−0.847 − 0.530i)17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0620i)2-s + (−0.806 − 0.591i)3-s + (0.992 + 0.123i)4-s + (−0.972 − 0.233i)5-s + (0.767 + 0.640i)6-s + (−0.556 + 0.831i)7-s + (−0.982 − 0.185i)8-s + (0.299 + 0.954i)9-s + (0.955 + 0.293i)10-s + (−0.635 + 0.771i)11-s + (−0.726 − 0.687i)12-s + (0.664 + 0.747i)13-s + (0.606 − 0.794i)14-s + (0.645 + 0.763i)15-s + (0.969 + 0.245i)16-s + (−0.847 − 0.530i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.826 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.826 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.826 - 0.562i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.826 - 0.562i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01749743301 - 0.05680045154i\)
\(L(\frac12)\) \(\approx\) \(0.01749743301 - 0.05680045154i\)
\(L(1)\) \(\approx\) \(0.3517992271 + 0.01161540043i\)
\(L(1)\) \(\approx\) \(0.3517992271 + 0.01161540043i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.998 - 0.0620i)T \)
3 \( 1 + (-0.806 - 0.591i)T \)
5 \( 1 + (-0.972 - 0.233i)T \)
7 \( 1 + (-0.556 + 0.831i)T \)
11 \( 1 + (-0.635 + 0.771i)T \)
13 \( 1 + (0.664 + 0.747i)T \)
17 \( 1 + (-0.847 - 0.530i)T \)
19 \( 1 + (0.0806 + 0.996i)T \)
29 \( 1 + (-0.673 - 0.739i)T \)
31 \( 1 + (0.879 - 0.476i)T \)
37 \( 1 + (-0.999 - 0.0372i)T \)
41 \( 1 + (0.751 - 0.659i)T \)
43 \( 1 + (-0.996 - 0.0868i)T \)
47 \( 1 + (-0.0682 + 0.997i)T \)
53 \( 1 + (0.955 - 0.293i)T \)
59 \( 1 + (-0.358 + 0.933i)T \)
61 \( 1 + (-0.616 + 0.787i)T \)
67 \( 1 + (0.392 - 0.919i)T \)
71 \( 1 + (0.275 + 0.961i)T \)
73 \( 1 + (-0.673 + 0.739i)T \)
79 \( 1 + (0.227 - 0.973i)T \)
83 \( 1 + (0.369 - 0.929i)T \)
89 \( 1 + (-0.896 + 0.443i)T \)
97 \( 1 + (0.545 - 0.837i)T \)
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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

−23.71341215096343471464257037839, −23.17666745651474593923409520160, −22.18409309596379329060070766395, −21.20372983922629398526862987399, −20.199804754362743087797472682341, −19.6837352991939295981004465176, −18.642564947153458639844491700441, −17.88331512458458121115371570811, −17.00997062374424818129922489622, −16.167486875574179911649851428441, −15.692004688462119495680378641877, −15.018894604392548883476775455565, −13.39553802963058596155259012749, −12.364683916689873787654289119759, −11.17827795025989842643910644924, −10.8670691295136049125455807151, −10.16593210157468981108513157677, −8.94038199218075945297276422225, −8.09715354677832348245499771407, −6.98390661950266602691954962579, −6.35892905566330678626068887595, −5.11613234817358538429417194912, −3.759057840008765902678738017, −3.02490019399557734515439745110, −0.931864858015726205052755202309, 0.06381286202435737526873346857, 1.617014496644764749557502756086, 2.64503930689721721764183655234, 4.16057284692728345737623140821, 5.514114272649851381239397151939, 6.48921132315013362880796805387, 7.29242063461288500730260951877, 8.13691214017003416981495473975, 9.0427319718299880624846043652, 10.11533706754724423246820571048, 11.13276695865115949296351753845, 11.874324530782313940850689293822, 12.374001467970420163867353526177, 13.374643713795268281185519000520, 15.10234649638486702572085013934, 15.853570374908867510624225669227, 16.32097807679573652314858945456, 17.32571395468470343866075930006, 18.30403488483392250344237350288, 18.7835771926442640659540860394, 19.41734438028788078020586432050, 20.462597248598712928521072792387, 21.27941710306201175064297184441, 22.62093115484375253157485091613, 23.08448487829362111100341153918

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