Properties

Label 2-531-1.1-c7-0-57
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  − 5.01·2-s − 102.·4-s + 77.5·5-s − 216.·7-s + 1.15e3·8-s − 389.·10-s − 3.37e3·11-s + 1.24e4·13-s + 1.08e3·14-s + 7.34e3·16-s + 2.63e4·17-s + 4.14e4·19-s − 7.97e3·20-s + 1.69e4·22-s + 4.21e4·23-s − 7.21e4·25-s − 6.24e4·26-s + 2.22e4·28-s + 1.51e5·29-s − 2.39e5·31-s − 1.85e5·32-s − 1.32e5·34-s − 1.68e4·35-s − 4.11e5·37-s − 2.07e5·38-s + 8.98e4·40-s + 5.36e5·41-s + ⋯
L(s)  = 1  − 0.443·2-s − 0.803·4-s + 0.277·5-s − 0.238·7-s + 0.799·8-s − 0.123·10-s − 0.764·11-s + 1.57·13-s + 0.105·14-s + 0.448·16-s + 1.30·17-s + 1.38·19-s − 0.222·20-s + 0.339·22-s + 0.721·23-s − 0.922·25-s − 0.697·26-s + 0.191·28-s + 1.15·29-s − 1.44·31-s − 0.998·32-s − 0.577·34-s − 0.0663·35-s − 1.33·37-s − 0.614·38-s + 0.221·40-s + 1.21·41-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

\(L(4)\) \(\approx\) \(1.680246177\)
\(L(\frac12)\) \(\approx\) \(1.680246177\)
\(L(\frac{9}{2})\) 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 \( 1 + 5.01T + 128T^{2} \)
5 \( 1 - 77.5T + 7.81e4T^{2} \)
7 \( 1 + 216.T + 8.23e5T^{2} \)
11 \( 1 + 3.37e3T + 1.94e7T^{2} \)
13 \( 1 - 1.24e4T + 6.27e7T^{2} \)
17 \( 1 - 2.63e4T + 4.10e8T^{2} \)
19 \( 1 - 4.14e4T + 8.93e8T^{2} \)
23 \( 1 - 4.21e4T + 3.40e9T^{2} \)
29 \( 1 - 1.51e5T + 1.72e10T^{2} \)
31 \( 1 + 2.39e5T + 2.75e10T^{2} \)
37 \( 1 + 4.11e5T + 9.49e10T^{2} \)
41 \( 1 - 5.36e5T + 1.94e11T^{2} \)
43 \( 1 - 5.83e5T + 2.71e11T^{2} \)
47 \( 1 - 5.87e5T + 5.06e11T^{2} \)
53 \( 1 - 2.67e5T + 1.17e12T^{2} \)
61 \( 1 + 4.98e5T + 3.14e12T^{2} \)
67 \( 1 - 2.87e6T + 6.06e12T^{2} \)
71 \( 1 + 1.08e6T + 9.09e12T^{2} \)
73 \( 1 + 4.74e3T + 1.10e13T^{2} \)
79 \( 1 + 1.41e6T + 1.92e13T^{2} \)
83 \( 1 + 3.72e6T + 2.71e13T^{2} \)
89 \( 1 + 1.04e7T + 4.42e13T^{2} \)
97 \( 1 + 4.51e6T + 8.07e13T^{2} \)
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

−9.642121908941079998388424305824, −8.898040102986413872558202650747, −8.026225467699063796620126424344, −7.27226454344081872186240935448, −5.80431987611960109105953883367, −5.28334299108931074265540307451, −3.92995609672327441075560046327, −3.08706475874166354844447551292, −1.44886491987207639609741360766, −0.67538919703999010564868657626, 0.67538919703999010564868657626, 1.44886491987207639609741360766, 3.08706475874166354844447551292, 3.92995609672327441075560046327, 5.28334299108931074265540307451, 5.80431987611960109105953883367, 7.27226454344081872186240935448, 8.026225467699063796620126424344, 8.898040102986413872558202650747, 9.642121908941079998388424305824

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