Properties

Label 2-531-59.58-c4-0-86
Degree $2$
Conductor $531$
Sign $-0.380 + 0.924i$
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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3.44i·2-s + 4.13·4-s + 16.2·5-s + 92.6·7-s − 69.3i·8-s − 55.8i·10-s + 48.6i·11-s − 216. i·13-s − 319. i·14-s − 172.·16-s − 171.·17-s + 267.·19-s + 67.1·20-s + 167.·22-s − 734. i·23-s + ⋯
L(s)  = 1  − 0.861i·2-s + 0.258·4-s + 0.648·5-s + 1.89·7-s − 1.08i·8-s − 0.558i·10-s + 0.402i·11-s − 1.28i·13-s − 1.62i·14-s − 0.674·16-s − 0.595·17-s + 0.740·19-s + 0.167·20-s + 0.346·22-s − 1.38i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $-0.380 + 0.924i$
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: \((2,\ 531,\ (\ :2),\ -0.380 + 0.924i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.489209720\)
\(L(\frac12)\) \(\approx\) \(3.489209720\)
\(L(3)\) 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 \( 1 + (-1.32e3 + 3.21e3i)T \)
good2 \( 1 + 3.44iT - 16T^{2} \)
5 \( 1 - 16.2T + 625T^{2} \)
7 \( 1 - 92.6T + 2.40e3T^{2} \)
11 \( 1 - 48.6iT - 1.46e4T^{2} \)
13 \( 1 + 216. iT - 2.85e4T^{2} \)
17 \( 1 + 171.T + 8.35e4T^{2} \)
19 \( 1 - 267.T + 1.30e5T^{2} \)
23 \( 1 + 734. iT - 2.79e5T^{2} \)
29 \( 1 + 999.T + 7.07e5T^{2} \)
31 \( 1 + 568. iT - 9.23e5T^{2} \)
37 \( 1 - 1.00e3iT - 1.87e6T^{2} \)
41 \( 1 - 2.30e3T + 2.82e6T^{2} \)
43 \( 1 - 2.15e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.19e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.10e3T + 7.89e6T^{2} \)
61 \( 1 - 4.03e3iT - 1.38e7T^{2} \)
67 \( 1 + 8.14e3iT - 2.01e7T^{2} \)
71 \( 1 - 2.53e3T + 2.54e7T^{2} \)
73 \( 1 + 792. iT - 2.83e7T^{2} \)
79 \( 1 + 3.21e3T + 3.89e7T^{2} \)
83 \( 1 - 1.05e4iT - 4.74e7T^{2} \)
89 \( 1 - 226. iT - 6.27e7T^{2} \)
97 \( 1 + 7.28e3iT - 8.85e7T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.20333412550284331268941585900, −9.333169687115794266050627644320, −8.103664384976849100058249563034, −7.47306937944569715885567530633, −6.14789945576178821434983772029, −5.14367457458397358127370670524, −4.15585399921826209793011374346, −2.65614198933273618515107589392, −1.87690960279627635128427612282, −0.878728081671032775139294967070, 1.54615409831733517586044600741, 2.18719101012990617658983144286, 4.09746121382433719345810568930, 5.29858011149401130955724839234, 5.75109945302540882998693409703, 7.07122069806491633131062146205, 7.62096897953862740964824278298, 8.616888110641147892104535316826, 9.329298824177644435672167602342, 10.72035739818541807279512360746

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