Properties

Label 1-23e2-529.8-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.431 + 0.902i$
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  + (0.566 + 0.824i)2-s + (0.984 − 0.172i)3-s + (−0.358 + 0.933i)4-s + (0.392 + 0.919i)5-s + (0.700 + 0.713i)6-s + (0.975 − 0.221i)7-s + (−0.972 + 0.233i)8-s + (0.940 − 0.340i)9-s + (−0.535 + 0.844i)10-s + (−0.847 + 0.530i)11-s + (−0.191 + 0.981i)12-s + (−0.287 + 0.957i)13-s + (0.735 + 0.678i)14-s + (0.545 + 0.837i)15-s + (−0.743 − 0.668i)16-s + (−0.759 + 0.650i)17-s + ⋯
L(s)  = 1  + (0.566 + 0.824i)2-s + (0.984 − 0.172i)3-s + (−0.358 + 0.933i)4-s + (0.392 + 0.919i)5-s + (0.700 + 0.713i)6-s + (0.975 − 0.221i)7-s + (−0.972 + 0.233i)8-s + (0.940 − 0.340i)9-s + (−0.535 + 0.844i)10-s + (−0.847 + 0.530i)11-s + (−0.191 + 0.981i)12-s + (−0.287 + 0.957i)13-s + (0.735 + 0.678i)14-s + (0.545 + 0.837i)15-s + (−0.743 − 0.668i)16-s + (−0.759 + 0.650i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.431 + 0.902i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.431 + 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.426483641 + 2.263738638i\)
\(L(\frac12)\) \(\approx\) \(1.426483641 + 2.263738638i\)
\(L(1)\) \(\approx\) \(1.553773980 + 1.188639060i\)
\(L(1)\) \(\approx\) \(1.553773980 + 1.188639060i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.566 + 0.824i)T \)
3 \( 1 + (0.984 - 0.172i)T \)
5 \( 1 + (0.392 + 0.919i)T \)
7 \( 1 + (0.975 - 0.221i)T \)
11 \( 1 + (-0.847 + 0.530i)T \)
13 \( 1 + (-0.287 + 0.957i)T \)
17 \( 1 + (-0.759 + 0.650i)T \)
19 \( 1 + (-0.311 - 0.950i)T \)
29 \( 1 + (0.664 - 0.747i)T \)
31 \( 1 + (0.105 - 0.994i)T \)
37 \( 1 + (-0.263 + 0.964i)T \)
41 \( 1 + (0.997 + 0.0744i)T \)
43 \( 1 + (0.995 + 0.0991i)T \)
47 \( 1 + (0.682 - 0.730i)T \)
53 \( 1 + (-0.535 - 0.844i)T \)
59 \( 1 + (0.251 + 0.967i)T \)
61 \( 1 + (-0.952 - 0.305i)T \)
67 \( 1 + (-0.999 - 0.0124i)T \)
71 \( 1 + (0.346 - 0.938i)T \)
73 \( 1 + (0.664 + 0.747i)T \)
79 \( 1 + (-0.982 + 0.185i)T \)
83 \( 1 + (-0.635 - 0.771i)T \)
89 \( 1 + (0.931 + 0.363i)T \)
97 \( 1 + (-0.977 - 0.209i)T \)
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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

−23.2379699562378511462691088844, −21.98482383690246102293042533844, −21.260250563891798887648151309478, −20.76052723120897385468110675022, −20.170511869993528008644668795694, −19.32421970926854789353226440244, −18.262845911232029440589094979034, −17.65058026765324140026203252950, −16.04700721629244746102022701706, −15.438177818874508565680615428195, −14.26351456825707150013112933494, −13.90540976851463038971035735382, −12.7898168617058545582780944114, −12.38729378982262082755623568479, −10.923786005546533683189817712778, −10.30536069833814125484762627166, −9.16497575447278164537750504721, −8.54896612854214242837777718939, −7.62910528827051184754247243299, −5.80625408186134121579448058865, −5.005714707587065924957724093239, −4.284604160307078191865948647990, −2.96756809879051652462598871676, −2.173349068421232071365528268064, −1.11013343788601444432798106138, 2.080996118703702938773790139895, 2.67203461783552966985707876466, 4.09635828600026885910604305604, 4.71881631845068742447938603672, 6.17623256852458248794439811634, 7.07402276151779800108025858578, 7.69114951229535121850951468093, 8.60739005211589979371846609126, 9.58808033468540178712880861291, 10.745091975722520611658337313087, 11.83004809560688826335304371410, 13.08649511582925181876528117077, 13.679347877963082616671659750123, 14.40708380891195615338898701856, 15.1318927866147698577376420790, 15.600769153835216052666241198399, 17.12321035955390377503132489399, 17.791213525996460616110936368317, 18.51454513938463701161509009470, 19.508279619454138581136524735573, 20.74632462579523465963945935369, 21.309301638738447511663389447795, 21.971562107865473611316593737420, 23.0906291578392791053315055907, 24.061786720268199441126693972392

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