Properties

Label 2-531-59.58-c4-0-34
Degree $2$
Conductor $531$
Sign $0.0950 + 0.995i$
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  − 4.96i·2-s − 8.66·4-s − 41.3·5-s + 1.08·7-s − 36.4i·8-s + 205. i·10-s + 142. i·11-s + 35.2i·13-s − 5.39i·14-s − 319.·16-s − 527.·17-s + 76.0·19-s + 358.·20-s + 708.·22-s − 138. i·23-s + ⋯
L(s)  = 1  − 1.24i·2-s − 0.541·4-s − 1.65·5-s + 0.0221·7-s − 0.569i·8-s + 2.05i·10-s + 1.17i·11-s + 0.208i·13-s − 0.0275i·14-s − 1.24·16-s − 1.82·17-s + 0.210·19-s + 0.897·20-s + 1.46·22-s − 0.262i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0950 + 0.995i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0950 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $0.0950 + 0.995i$
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: \((2,\ 531,\ (\ :2),\ 0.0950 + 0.995i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.041193801\)
\(L(\frac12)\) \(\approx\) \(1.041193801\)
\(L(3)\) 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 \( 1 + (330. + 3.46e3i)T \)
good2 \( 1 + 4.96iT - 16T^{2} \)
5 \( 1 + 41.3T + 625T^{2} \)
7 \( 1 - 1.08T + 2.40e3T^{2} \)
11 \( 1 - 142. iT - 1.46e4T^{2} \)
13 \( 1 - 35.2iT - 2.85e4T^{2} \)
17 \( 1 + 527.T + 8.35e4T^{2} \)
19 \( 1 - 76.0T + 1.30e5T^{2} \)
23 \( 1 + 138. iT - 2.79e5T^{2} \)
29 \( 1 - 1.04e3T + 7.07e5T^{2} \)
31 \( 1 - 1.36e3iT - 9.23e5T^{2} \)
37 \( 1 + 2.07e3iT - 1.87e6T^{2} \)
41 \( 1 - 1.60e3T + 2.82e6T^{2} \)
43 \( 1 - 746. iT - 3.41e6T^{2} \)
47 \( 1 - 613. iT - 4.87e6T^{2} \)
53 \( 1 - 3.30e3T + 7.89e6T^{2} \)
61 \( 1 + 3.02e3iT - 1.38e7T^{2} \)
67 \( 1 - 3.67e3iT - 2.01e7T^{2} \)
71 \( 1 + 1.82e3T + 2.54e7T^{2} \)
73 \( 1 - 2.69e3iT - 2.83e7T^{2} \)
79 \( 1 - 3.08e3T + 3.89e7T^{2} \)
83 \( 1 + 7.95e3iT - 4.74e7T^{2} \)
89 \( 1 - 5.14e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.30e4iT - 8.85e7T^{2} \)
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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

−10.35483619249186139314391453717, −9.269797790394506940056683370696, −8.434755854366738069024821985969, −7.26476652159373395479634233668, −6.71190951830077900521879187534, −4.61940621563684592771913702793, −4.22219343308693966850527432903, −3.08899869873966879742008410400, −1.99136752446078903878055138148, −0.54839391185612415300572686677, 0.52891481922997491268863271095, 2.72942947521523359136221937524, 4.00979867837455175509203473646, 4.86539266936343995170832990379, 6.10860926669132118620474712262, 6.86619578189944252795825368346, 7.78947960916367021629753616128, 8.344895431255666587835778890396, 9.016914632853652672815988012272, 10.67659498485399410394987487360

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