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Properties

Label 1-23e2-529.349-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.391 - 0.920i$
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  + (0.606 + 0.794i)2-s + (−0.358 − 0.933i)3-s + (−0.263 + 0.964i)4-s + (−0.616 + 0.787i)5-s + (0.524 − 0.851i)6-s + (0.767 − 0.640i)7-s + (−0.926 + 0.375i)8-s + (−0.743 + 0.668i)9-s + (−0.999 − 0.0124i)10-s + (−0.166 − 0.985i)11-s + (0.995 − 0.0991i)12-s + (−0.847 − 0.530i)13-s + (0.975 + 0.221i)14-s + (0.955 + 0.293i)15-s + (−0.860 − 0.508i)16-s + (−0.404 + 0.914i)17-s + ⋯
L(s)  = 1  + (0.606 + 0.794i)2-s + (−0.358 − 0.933i)3-s + (−0.263 + 0.964i)4-s + (−0.616 + 0.787i)5-s + (0.524 − 0.851i)6-s + (0.767 − 0.640i)7-s + (−0.926 + 0.375i)8-s + (−0.743 + 0.668i)9-s + (−0.999 − 0.0124i)10-s + (−0.166 − 0.985i)11-s + (0.995 − 0.0991i)12-s + (−0.847 − 0.530i)13-s + (0.975 + 0.221i)14-s + (0.955 + 0.293i)15-s + (−0.860 − 0.508i)16-s + (−0.404 + 0.914i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.391 - 0.920i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.391 - 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1976915423 - 0.2988477715i\)
\(L(\frac12)\) \(\approx\) \(0.1976915423 - 0.2988477715i\)
\(L(1)\) \(\approx\) \(0.8165678577 + 0.1200245739i\)
\(L(1)\) \(\approx\) \(0.8165678577 + 0.1200245739i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.606 + 0.794i)T \)
3 \( 1 + (-0.358 - 0.933i)T \)
5 \( 1 + (-0.616 + 0.787i)T \)
7 \( 1 + (0.767 - 0.640i)T \)
11 \( 1 + (-0.166 - 0.985i)T \)
13 \( 1 + (-0.847 - 0.530i)T \)
17 \( 1 + (-0.404 + 0.914i)T \)
19 \( 1 + (-0.998 - 0.0620i)T \)
29 \( 1 + (-0.635 + 0.771i)T \)
31 \( 1 + (0.481 - 0.876i)T \)
37 \( 1 + (-0.381 - 0.924i)T \)
41 \( 1 + (-0.287 - 0.957i)T \)
43 \( 1 + (-0.791 - 0.611i)T \)
47 \( 1 + (-0.0682 + 0.997i)T \)
53 \( 1 + (-0.999 + 0.0124i)T \)
59 \( 1 + (0.0806 - 0.996i)T \)
61 \( 1 + (-0.0929 + 0.995i)T \)
67 \( 1 + (-0.952 + 0.305i)T \)
71 \( 1 + (0.545 - 0.837i)T \)
73 \( 1 + (-0.635 - 0.771i)T \)
79 \( 1 + (0.0558 + 0.998i)T \)
83 \( 1 + (0.998 - 0.0496i)T \)
89 \( 1 + (-0.993 - 0.111i)T \)
97 \( 1 + (-0.535 - 0.844i)T \)
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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

−23.44378158430194257407006294374, −22.87503220565119729592838835582, −21.87894591046381505929896902325, −21.24831684065908183475518872344, −20.52866656716846029881775422445, −19.95218572402937488225319899637, −18.90755301664484086841184149733, −17.817270500289304347100128066048, −16.97806468592396590113872014283, −15.834706184473456191794706367089, −15.09530578564827033285135401181, −14.62379575414597585236456288976, −13.297545663652340944807982783905, −12.097081326213134379377847708484, −11.889218939435873342225750704986, −10.97531365100715841356595825421, −9.85713186046533453562252072547, −9.20332717523356403390497068494, −8.23482580584767625565068333001, −6.668978886853627085519432271623, −5.272788765931573689428909461998, −4.7411379572576086252930513695, −4.23409374536579317555178380632, −2.82234576425691544598670337327, −1.687309507613584525173088732639, 0.15276210181942675264395020611, 2.14089627935822686919244774353, 3.31415822744065986165517699194, 4.38317590773015825017829365743, 5.51706470628344563560881504743, 6.419167665527968507684274615243, 7.257787392674112841291559085538, 7.92200599516698297779135975178, 8.58637140514242672732222298175, 10.64012392301201364643593470942, 11.1835557054033353065708722637, 12.18750728270154642399457172275, 13.01393191896126005790201859410, 13.9097909586886969012499935205, 14.61230644284208739231643420736, 15.34125489923659893821889335345, 16.58276418362029458348851783771, 17.26786500688947739337542463447, 17.9322965569384217990534279429, 18.929303889276570360214640968743, 19.657495638564658180139860925372, 20.86240305729121889835829772631, 22.02217355155770875459836443142, 22.46689858591983269932436900248, 23.62973085305867553035105530352

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