Properties

Label 2-531-59.58-c4-0-25
Degree $2$
Conductor $531$
Sign $-0.444 - 0.895i$
Analytic cond. $54.8894$
Root an. cond. $7.40874$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.77i·2-s − 17.3·4-s − 34.3·5-s − 39.2·7-s − 7.56i·8-s − 198. i·10-s − 56.8i·11-s − 232. i·13-s − 226. i·14-s − 233.·16-s + 391.·17-s − 352.·19-s + 594.·20-s + 328.·22-s + 173. i·23-s + ⋯
L(s)  = 1  + 1.44i·2-s − 1.08·4-s − 1.37·5-s − 0.801·7-s − 0.118i·8-s − 1.98i·10-s − 0.469i·11-s − 1.37i·13-s − 1.15i·14-s − 0.911·16-s + 1.35·17-s − 0.976·19-s + 1.48·20-s + 0.677·22-s + 0.327i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $-0.444 - 0.895i$
Analytic conductor: \(54.8894\)
Root analytic conductor: \(7.40874\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{531} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :2),\ -0.444 - 0.895i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.8653730655\)
\(L(\frac12)\) \(\approx\) \(0.8653730655\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 + (-1.54e3 - 3.11e3i)T \)
good2 \( 1 - 5.77iT - 16T^{2} \)
5 \( 1 + 34.3T + 625T^{2} \)
7 \( 1 + 39.2T + 2.40e3T^{2} \)
11 \( 1 + 56.8iT - 1.46e4T^{2} \)
13 \( 1 + 232. iT - 2.85e4T^{2} \)
17 \( 1 - 391.T + 8.35e4T^{2} \)
19 \( 1 + 352.T + 1.30e5T^{2} \)
23 \( 1 - 173. iT - 2.79e5T^{2} \)
29 \( 1 + 90.7T + 7.07e5T^{2} \)
31 \( 1 - 128. iT - 9.23e5T^{2} \)
37 \( 1 + 413. iT - 1.87e6T^{2} \)
41 \( 1 - 761.T + 2.82e6T^{2} \)
43 \( 1 - 874. iT - 3.41e6T^{2} \)
47 \( 1 + 895. iT - 4.87e6T^{2} \)
53 \( 1 + 3.68e3T + 7.89e6T^{2} \)
61 \( 1 - 1.49e3iT - 1.38e7T^{2} \)
67 \( 1 + 3.68e3iT - 2.01e7T^{2} \)
71 \( 1 - 9.23e3T + 2.54e7T^{2} \)
73 \( 1 - 4.30e3iT - 2.83e7T^{2} \)
79 \( 1 + 4.70e3T + 3.89e7T^{2} \)
83 \( 1 - 398. iT - 4.74e7T^{2} \)
89 \( 1 - 1.24e4iT - 6.27e7T^{2} \)
97 \( 1 - 397. iT - 8.85e7T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.48011811031733148120328794605, −9.341330281463672888115498898550, −8.131721564528317582275096107041, −7.963893997203971791185078124764, −6.98462439807403582953213065565, −6.04245166354146505590804899819, −5.22092225597115980434623048742, −3.94100522035259441883242625511, −3.02797773200840488285618280250, −0.54940282005125941819276914459, 0.44513805285592445477714140704, 1.84198149119784282740384671471, 3.12467165263479809448324491543, 3.89893668712345492369787419523, 4.62691385213680540385224163276, 6.41429509906248093289086099107, 7.29302131644163749341991603139, 8.382592043272924879558232063794, 9.411250839539346327222017869092, 10.04569443848631970577121168567

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