Properties

Label 2-531-59.58-c4-0-58
Degree $2$
Conductor $531$
Sign $0.951 + 0.308i$
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  − 2.31i·2-s + 10.6·4-s + 38.6·5-s + 9.48·7-s − 61.7i·8-s − 89.6i·10-s + 58.4i·11-s + 290. i·13-s − 22.0i·14-s + 26.7·16-s + 467.·17-s − 261.·19-s + 410.·20-s + 135.·22-s − 190. i·23-s + ⋯
L(s)  = 1  − 0.579i·2-s + 0.663·4-s + 1.54·5-s + 0.193·7-s − 0.964i·8-s − 0.896i·10-s + 0.483i·11-s + 1.72i·13-s − 0.112i·14-s + 0.104·16-s + 1.61·17-s − 0.723·19-s + 1.02·20-s + 0.280·22-s − 0.359i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $0.951 + 0.308i$
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: \((2,\ 531,\ (\ :2),\ 0.951 + 0.308i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.863716956\)
\(L(\frac12)\) \(\approx\) \(3.863716956\)
\(L(3)\) 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 \( 1 + (3.31e3 + 1.07e3i)T \)
good2 \( 1 + 2.31iT - 16T^{2} \)
5 \( 1 - 38.6T + 625T^{2} \)
7 \( 1 - 9.48T + 2.40e3T^{2} \)
11 \( 1 - 58.4iT - 1.46e4T^{2} \)
13 \( 1 - 290. iT - 2.85e4T^{2} \)
17 \( 1 - 467.T + 8.35e4T^{2} \)
19 \( 1 + 261.T + 1.30e5T^{2} \)
23 \( 1 + 190. iT - 2.79e5T^{2} \)
29 \( 1 - 407.T + 7.07e5T^{2} \)
31 \( 1 - 1.89e3iT - 9.23e5T^{2} \)
37 \( 1 + 306. iT - 1.87e6T^{2} \)
41 \( 1 - 486.T + 2.82e6T^{2} \)
43 \( 1 + 1.83e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.57e3iT - 4.87e6T^{2} \)
53 \( 1 - 2.71e3T + 7.89e6T^{2} \)
61 \( 1 - 3.95e3iT - 1.38e7T^{2} \)
67 \( 1 - 5.26e3iT - 2.01e7T^{2} \)
71 \( 1 + 7.35e3T + 2.54e7T^{2} \)
73 \( 1 + 8.25e3iT - 2.83e7T^{2} \)
79 \( 1 - 8.38e3T + 3.89e7T^{2} \)
83 \( 1 - 8.21e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.36e4iT - 6.27e7T^{2} \)
97 \( 1 - 5.34e3iT - 8.85e7T^{2} \)
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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

−10.21547046278578146545973277220, −9.598720929097275153769373023063, −8.706028500375261650400061417806, −7.21866620803657608490710250839, −6.53595092512564413096816367024, −5.66606313354702869034561619783, −4.44942819036363574478997065771, −3.02038392954205591648558670253, −1.93387800601040989920786268738, −1.37700815784415476376801198187, 1.03436106545529497698746615611, 2.25917823681222760324090982381, 3.20466434724469373786217929845, 5.17832840096923824344174990584, 5.81313595037726580282041342437, 6.32517354583154268013196640137, 7.68961633765857129027016551102, 8.208084569871303485960319563867, 9.530245230139199702695112063373, 10.26502538586083288375441221658

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