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Properties

Label 1-23e2-529.463-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: $\chi_{529} (463, \cdot )$
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

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

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