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Properties

Label 1-23e2-529.254-r0-0-0
Degree $1$
Conductor $529$
Sign $0.999 + 0.0118i$
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  + (−0.0682 + 0.997i)2-s + (−0.0682 + 0.997i)3-s + (−0.990 − 0.136i)4-s + (−0.775 + 0.631i)5-s + (−0.990 − 0.136i)6-s + (−0.990 + 0.136i)7-s + (0.203 − 0.979i)8-s + (−0.990 − 0.136i)9-s + (−0.576 − 0.816i)10-s + (−0.334 − 0.942i)11-s + (0.203 − 0.979i)12-s + (−0.576 + 0.816i)13-s + (−0.0682 − 0.997i)14-s + (−0.576 − 0.816i)15-s + (0.962 + 0.269i)16-s + (−0.576 + 0.816i)17-s + ⋯
L(s)  = 1  + (−0.0682 + 0.997i)2-s + (−0.0682 + 0.997i)3-s + (−0.990 − 0.136i)4-s + (−0.775 + 0.631i)5-s + (−0.990 − 0.136i)6-s + (−0.990 + 0.136i)7-s + (0.203 − 0.979i)8-s + (−0.990 − 0.136i)9-s + (−0.576 − 0.816i)10-s + (−0.334 − 0.942i)11-s + (0.203 − 0.979i)12-s + (−0.576 + 0.816i)13-s + (−0.0682 − 0.997i)14-s + (−0.576 − 0.816i)15-s + (0.962 + 0.269i)16-s + (−0.576 + 0.816i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.999 + 0.0118i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (254, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.999 + 0.0118i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1062911978 + 0.0006312430472i\)
\(L(\frac12)\) \(\approx\) \(0.1062911978 + 0.0006312430472i\)
\(L(1)\) \(\approx\) \(0.3345356982 + 0.4067777731i\)
\(L(1)\) \(\approx\) \(0.3345356982 + 0.4067777731i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.0682 + 0.997i)T \)
3 \( 1 + (-0.0682 + 0.997i)T \)
5 \( 1 + (-0.775 + 0.631i)T \)
7 \( 1 + (-0.990 + 0.136i)T \)
11 \( 1 + (-0.334 - 0.942i)T \)
13 \( 1 + (-0.576 + 0.816i)T \)
17 \( 1 + (-0.576 + 0.816i)T \)
19 \( 1 + (-0.0682 + 0.997i)T \)
29 \( 1 + (-0.334 - 0.942i)T \)
31 \( 1 + (0.854 - 0.519i)T \)
37 \( 1 + (0.962 - 0.269i)T \)
41 \( 1 + (0.460 + 0.887i)T \)
43 \( 1 + (-0.917 - 0.398i)T \)
47 \( 1 + (0.854 + 0.519i)T \)
53 \( 1 + (-0.576 + 0.816i)T \)
59 \( 1 + (-0.0682 - 0.997i)T \)
61 \( 1 + (-0.775 + 0.631i)T \)
67 \( 1 + (-0.334 - 0.942i)T \)
71 \( 1 + (0.460 - 0.887i)T \)
73 \( 1 + (-0.334 + 0.942i)T \)
79 \( 1 + (-0.917 - 0.398i)T \)
83 \( 1 + (-0.775 + 0.631i)T \)
89 \( 1 + (0.682 - 0.730i)T \)
97 \( 1 + (0.460 - 0.887i)T \)
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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

−23.23030414462318789526977473200, −22.78606576770885422450399522331, −21.969631383303210249405163119142, −20.453509251292486597702060359994, −20.01520179642426860552352302440, −19.535143483508367214398204620223, −18.60287440062696643102487816867, −17.76620244491984258828277000258, −17.04771221955507326736468941217, −15.89610063049233462255340443392, −14.848302815547401890604239302535, −13.5053748958874836710951276771, −12.95672042126438194166479187198, −12.365184618708467241149226579261, −11.64842508257287362028169282114, −10.60315328735447664984813859248, −9.50138464793797338750965297782, −8.69732205077740801531281999176, −7.63694500573163624736817483088, −6.918723518676018272464219593969, −5.34840353662629771135257046077, −4.526291146312224198946637539190, −3.15600947347382802409507599077, −2.42408497419762618409398759505, −0.96100435369447015420176330769, 0.07391322953625909389781967221, 2.83854655954951472291998015395, 3.83531014101117751894592969745, 4.46438549568281749910109433565, 5.9496915579543543263735349187, 6.34722864110404615800230047108, 7.67813496927433484933144111167, 8.49740137269530995920461456396, 9.48121967651367403869833250312, 10.22389257427624241578942023260, 11.196112650574475484747976505730, 12.29309190702853924492280111306, 13.517210531128560761325945970491, 14.406329193466461390274815605686, 15.203643661780964719983694044692, 15.793326687659760528862734405857, 16.54661432191908460582856085339, 17.11200910570994111796101113636, 18.552657932840100855515361015942, 19.11135717331233045401616001313, 19.88821872885897517871772122721, 21.404519377996403070390338541024, 21.97806614129946717329418135680, 22.731963004413609435480206297641, 23.35537886409787228376755671044

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