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Properties

Label 1-23e2-529.369-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.433 + 0.901i$
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  + (−0.576 − 0.816i)2-s + (−0.576 − 0.816i)3-s + (−0.334 + 0.942i)4-s + (−0.990 + 0.136i)5-s + (−0.334 + 0.942i)6-s + (−0.334 − 0.942i)7-s + (0.962 − 0.269i)8-s + (−0.334 + 0.942i)9-s + (0.682 + 0.730i)10-s + (−0.0682 − 0.997i)11-s + (0.962 − 0.269i)12-s + (0.682 − 0.730i)13-s + (−0.576 + 0.816i)14-s + (0.682 + 0.730i)15-s + (−0.775 − 0.631i)16-s + (0.682 − 0.730i)17-s + ⋯
L(s)  = 1  + (−0.576 − 0.816i)2-s + (−0.576 − 0.816i)3-s + (−0.334 + 0.942i)4-s + (−0.990 + 0.136i)5-s + (−0.334 + 0.942i)6-s + (−0.334 − 0.942i)7-s + (0.962 − 0.269i)8-s + (−0.334 + 0.942i)9-s + (0.682 + 0.730i)10-s + (−0.0682 − 0.997i)11-s + (0.962 − 0.269i)12-s + (0.682 − 0.730i)13-s + (−0.576 + 0.816i)14-s + (0.682 + 0.730i)15-s + (−0.775 − 0.631i)16-s + (0.682 − 0.730i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.433 + 0.901i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.433 + 0.901i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2017649586 - 0.3209559591i\)
\(L(\frac12)\) \(\approx\) \(-0.2017649586 - 0.3209559591i\)
\(L(1)\) \(\approx\) \(0.3011641948 - 0.3890241878i\)
\(L(1)\) \(\approx\) \(0.3011641948 - 0.3890241878i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.576 - 0.816i)T \)
3 \( 1 + (-0.576 - 0.816i)T \)
5 \( 1 + (-0.990 + 0.136i)T \)
7 \( 1 + (-0.334 - 0.942i)T \)
11 \( 1 + (-0.0682 - 0.997i)T \)
13 \( 1 + (0.682 - 0.730i)T \)
17 \( 1 + (0.682 - 0.730i)T \)
19 \( 1 + (-0.576 - 0.816i)T \)
29 \( 1 + (-0.0682 - 0.997i)T \)
31 \( 1 + (0.203 - 0.979i)T \)
37 \( 1 + (-0.775 + 0.631i)T \)
41 \( 1 + (-0.917 + 0.398i)T \)
43 \( 1 + (0.854 - 0.519i)T \)
47 \( 1 + (0.203 + 0.979i)T \)
53 \( 1 + (0.682 - 0.730i)T \)
59 \( 1 + (-0.576 + 0.816i)T \)
61 \( 1 + (-0.990 + 0.136i)T \)
67 \( 1 + (-0.0682 - 0.997i)T \)
71 \( 1 + (-0.917 - 0.398i)T \)
73 \( 1 + (-0.0682 + 0.997i)T \)
79 \( 1 + (0.854 - 0.519i)T \)
83 \( 1 + (-0.990 + 0.136i)T \)
89 \( 1 + (0.460 + 0.887i)T \)
97 \( 1 + (-0.917 - 0.398i)T \)
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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

−23.82880419926408155013189183837, −23.27036391105682901565466866455, −22.72157016123442987215378771968, −21.64805264832704155199835944441, −20.68802048893613951097974465050, −19.6680775823599218685349006244, −18.820264758316060351775416000860, −18.11843055734969478354210445838, −17.03970051530299373615596320935, −16.31895226393063400421591042675, −15.69297729557109027233198936963, −15.04449106482222401665613211821, −14.35850172977824822329141196838, −12.58801458452483323106267744342, −11.988934179340244157062444412822, −10.82185681519794079835213746042, −10.10816131361871477429531755177, −8.98893112733572718872421144149, −8.519450044124971445551721099244, −7.232802573138407712125359897529, −6.30455086013101972278003852159, −5.40005079066223636694348160914, −4.4703230629266072993822556561, −3.52887494469314653425017082506, −1.58603588192075268432336643937, 0.34124575148318790856320431169, 1.04152391511699655091682615779, 2.74069775629798361015107085009, 3.576558171362752038332511517264, 4.69080739642838655487175731100, 6.175132720403005490011764155862, 7.28469181983264747620461128060, 7.87855655597397728203409184814, 8.70655378024521374944216568764, 10.17885211592865291677711658790, 10.96956673231272469124889900660, 11.48058481150646763835547920331, 12.384140695586531255659770989752, 13.34300369688028939014071133875, 13.80560433645531036234668001503, 15.553080531685278724450735592703, 16.49234583912123823481704683855, 17.045380535255001314761842398523, 18.0578108153054442578639245256, 18.9202380981172882359526328130, 19.31479442409133794251431896218, 20.19116484438103435059796111401, 20.99098617477136033945423093378, 22.33864677877952272249898057405, 22.82078547985868755179251129166

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