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Properties

Label 1-23e2-529.29-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.999 + 0.0199i$
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  + (−0.239 + 0.970i)2-s + (−0.0434 − 0.999i)3-s + (−0.885 − 0.465i)4-s + (−0.287 + 0.957i)5-s + (0.980 + 0.197i)6-s + (0.0558 + 0.998i)7-s + (0.664 − 0.747i)8-s + (−0.996 + 0.0868i)9-s + (−0.860 − 0.508i)10-s + (0.798 + 0.601i)11-s + (−0.426 + 0.904i)12-s + (−0.449 + 0.893i)13-s + (−0.982 − 0.185i)14-s + (0.969 + 0.245i)15-s + (0.566 + 0.824i)16-s + (−0.820 − 0.571i)17-s + ⋯
L(s)  = 1  + (−0.239 + 0.970i)2-s + (−0.0434 − 0.999i)3-s + (−0.885 − 0.465i)4-s + (−0.287 + 0.957i)5-s + (0.980 + 0.197i)6-s + (0.0558 + 0.998i)7-s + (0.664 − 0.747i)8-s + (−0.996 + 0.0868i)9-s + (−0.860 − 0.508i)10-s + (0.798 + 0.601i)11-s + (−0.426 + 0.904i)12-s + (−0.449 + 0.893i)13-s + (−0.982 − 0.185i)14-s + (0.969 + 0.245i)15-s + (0.566 + 0.824i)16-s + (−0.820 − 0.571i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.999 + 0.0199i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.999 + 0.0199i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005177607542 + 0.5183718897i\)
\(L(\frac12)\) \(\approx\) \(0.005177607542 + 0.5183718897i\)
\(L(1)\) \(\approx\) \(0.5808498690 + 0.3542180211i\)
\(L(1)\) \(\approx\) \(0.5808498690 + 0.3542180211i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.239 + 0.970i)T \)
3 \( 1 + (-0.0434 - 0.999i)T \)
5 \( 1 + (-0.287 + 0.957i)T \)
7 \( 1 + (0.0558 + 0.998i)T \)
11 \( 1 + (0.798 + 0.601i)T \)
13 \( 1 + (-0.449 + 0.893i)T \)
17 \( 1 + (-0.820 - 0.571i)T \)
19 \( 1 + (0.890 - 0.454i)T \)
29 \( 1 + (-0.977 + 0.209i)T \)
31 \( 1 + (-0.358 - 0.933i)T \)
37 \( 1 + (-0.896 - 0.443i)T \)
41 \( 1 + (-0.0186 + 0.999i)T \)
43 \( 1 + (0.999 + 0.0248i)T \)
47 \( 1 + (0.203 + 0.979i)T \)
53 \( 1 + (-0.860 + 0.508i)T \)
59 \( 1 + (0.323 - 0.946i)T \)
61 \( 1 + (-0.759 + 0.650i)T \)
67 \( 1 + (-0.709 + 0.704i)T \)
71 \( 1 + (0.299 + 0.954i)T \)
73 \( 1 + (-0.977 - 0.209i)T \)
79 \( 1 + (-0.673 + 0.739i)T \)
83 \( 1 + (-0.535 - 0.844i)T \)
89 \( 1 + (-0.0929 + 0.995i)T \)
97 \( 1 + (-0.743 + 0.668i)T \)
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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

−22.62859093477283619928235725138, −22.23158471681927075095455011233, −21.13433083337736279308151057354, −20.48035307461320405120584192091, −19.911584077450076413253762822159, −19.35884453095731991190086757684, −17.77699489516832109012566567162, −17.08151556710448788018284179950, −16.57447011130414314042416803737, −15.53824948900519995670379704678, −14.311967051205469164358632792642, −13.567378926384360083983740989886, −12.54706928144445701088344247670, −11.65658054629217619817457531391, −10.829025459871143120874072358275, −10.10226921803164582273095494205, −9.16323160319985863378532741135, −8.513826775257910606069399701547, −7.50091924621144792873965362835, −5.65762156699079128575100440275, −4.75233445994637542209705532598, −3.88780367846948832023304705575, −3.305263208138549607245166202931, −1.561921128090690001946957674, −0.30885987276175170259512631317, 1.670890001110709045885163035, 2.7504762658892225394764529363, 4.27112907129238862207727536261, 5.51918127592353862704186808658, 6.44327912711054646789465468959, 7.09332400015927294299322472085, 7.72828387743804009875263770501, 9.02271502475928413496785631681, 9.4960123335540997884457300408, 11.1632381448640894862472262169, 11.81800831129372165060099324816, 12.85340385992760747828693516598, 13.97271043302768390919485337943, 14.52903300901807655504488994839, 15.29027973245506910237732178092, 16.24360955871722553158277410541, 17.440406902813616425888982652407, 17.93176433729680659773145800263, 18.78917816565467031462356115711, 19.2020901875655046920594811014, 20.20129762303126291043222597831, 22.05325581922200922989475770176, 22.32247902105576941061848903583, 23.1423958302404452823378395733, 24.2850397184932787349111273802

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