Properties

Label 2-531-1.1-c7-0-23
Degree $2$
Conductor $531$
Sign $1$
Analytic cond. $165.876$
Root an. cond. $12.8793$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 15.5·2-s + 113.·4-s + 141.·5-s + 105.·7-s + 232.·8-s − 2.20e3·10-s + 555.·11-s − 1.12e4·13-s − 1.63e3·14-s − 1.80e4·16-s + 3.13e3·17-s − 2.19e4·19-s + 1.60e4·20-s − 8.61e3·22-s − 2.85e4·23-s − 5.80e4·25-s + 1.75e5·26-s + 1.19e4·28-s − 1.60e5·29-s − 1.63e5·31-s + 2.50e5·32-s − 4.87e4·34-s + 1.49e4·35-s − 1.31e5·37-s + 3.40e5·38-s + 3.29e4·40-s + 2.77e5·41-s + ⋯
L(s)  = 1  − 1.37·2-s + 0.883·4-s + 0.507·5-s + 0.116·7-s + 0.160·8-s − 0.696·10-s + 0.125·11-s − 1.42·13-s − 0.159·14-s − 1.10·16-s + 0.155·17-s − 0.734·19-s + 0.448·20-s − 0.172·22-s − 0.489·23-s − 0.742·25-s + 1.95·26-s + 0.102·28-s − 1.22·29-s − 0.986·31-s + 1.35·32-s − 0.212·34-s + 0.0588·35-s − 0.425·37-s + 1.00·38-s + 0.0813·40-s + 0.627·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $1$
Analytic conductor: \(165.876\)
Root analytic conductor: \(12.8793\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{531} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(0.5696253529\)
\(L(\frac12)\) \(\approx\) \(0.5696253529\)
\(L(\frac{9}{2})\) 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 \( 1 - 2.05e5T \)
good2 \( 1 + 15.5T + 128T^{2} \)
5 \( 1 - 141.T + 7.81e4T^{2} \)
7 \( 1 - 105.T + 8.23e5T^{2} \)
11 \( 1 - 555.T + 1.94e7T^{2} \)
13 \( 1 + 1.12e4T + 6.27e7T^{2} \)
17 \( 1 - 3.13e3T + 4.10e8T^{2} \)
19 \( 1 + 2.19e4T + 8.93e8T^{2} \)
23 \( 1 + 2.85e4T + 3.40e9T^{2} \)
29 \( 1 + 1.60e5T + 1.72e10T^{2} \)
31 \( 1 + 1.63e5T + 2.75e10T^{2} \)
37 \( 1 + 1.31e5T + 9.49e10T^{2} \)
41 \( 1 - 2.77e5T + 1.94e11T^{2} \)
43 \( 1 - 6.21e5T + 2.71e11T^{2} \)
47 \( 1 - 1.39e6T + 5.06e11T^{2} \)
53 \( 1 + 1.49e6T + 1.17e12T^{2} \)
61 \( 1 + 2.41e6T + 3.14e12T^{2} \)
67 \( 1 - 2.34e6T + 6.06e12T^{2} \)
71 \( 1 + 5.27e4T + 9.09e12T^{2} \)
73 \( 1 - 1.12e6T + 1.10e13T^{2} \)
79 \( 1 - 3.33e6T + 1.92e13T^{2} \)
83 \( 1 - 5.69e6T + 2.71e13T^{2} \)
89 \( 1 - 5.02e6T + 4.42e13T^{2} \)
97 \( 1 - 9.04e6T + 8.07e13T^{2} \)
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

−9.430311192387847207122249393214, −9.180129549731568877724332482692, −7.86691718492362526756528563042, −7.43420240370458302971297097653, −6.29145064108581303146379289376, −5.18643501352540033073809496268, −4.03438673817452235742614820686, −2.37582853765647136029115556677, −1.72457213820629353152334023600, −0.39648726521804082899851102717, 0.39648726521804082899851102717, 1.72457213820629353152334023600, 2.37582853765647136029115556677, 4.03438673817452235742614820686, 5.18643501352540033073809496268, 6.29145064108581303146379289376, 7.43420240370458302971297097653, 7.86691718492362526756528563042, 9.180129549731568877724332482692, 9.430311192387847207122249393214

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