Properties

Label 2-531-1.1-c7-0-15
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  − 0.396·2-s − 127.·4-s − 247.·5-s − 652.·7-s + 101.·8-s + 98.1·10-s + 2.47e3·11-s + 9.41e3·13-s + 259.·14-s + 1.63e4·16-s − 1.25e4·17-s − 4.13e4·19-s + 3.16e4·20-s − 983.·22-s + 5.97e4·23-s − 1.70e4·25-s − 3.73e3·26-s + 8.34e4·28-s − 2.12e5·29-s − 7.79e3·31-s − 1.94e4·32-s + 4.99e3·34-s + 1.61e5·35-s − 4.47e5·37-s + 1.64e4·38-s − 2.51e4·40-s − 1.40e4·41-s + ⋯
L(s)  = 1  − 0.0350·2-s − 0.998·4-s − 0.884·5-s − 0.719·7-s + 0.0701·8-s + 0.0310·10-s + 0.561·11-s + 1.18·13-s + 0.0252·14-s + 0.996·16-s − 0.620·17-s − 1.38·19-s + 0.883·20-s − 0.0196·22-s + 1.02·23-s − 0.217·25-s − 0.0417·26-s + 0.718·28-s − 1.61·29-s − 0.0469·31-s − 0.105·32-s + 0.0217·34-s + 0.636·35-s − 1.45·37-s + 0.0485·38-s − 0.0620·40-s − 0.0317·41-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

\(L(4)\) \(\approx\) \(0.4190707820\)
\(L(\frac12)\) \(\approx\) \(0.4190707820\)
\(L(\frac{9}{2})\) 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 \( 1 + 0.396T + 128T^{2} \)
5 \( 1 + 247.T + 7.81e4T^{2} \)
7 \( 1 + 652.T + 8.23e5T^{2} \)
11 \( 1 - 2.47e3T + 1.94e7T^{2} \)
13 \( 1 - 9.41e3T + 6.27e7T^{2} \)
17 \( 1 + 1.25e4T + 4.10e8T^{2} \)
19 \( 1 + 4.13e4T + 8.93e8T^{2} \)
23 \( 1 - 5.97e4T + 3.40e9T^{2} \)
29 \( 1 + 2.12e5T + 1.72e10T^{2} \)
31 \( 1 + 7.79e3T + 2.75e10T^{2} \)
37 \( 1 + 4.47e5T + 9.49e10T^{2} \)
41 \( 1 + 1.40e4T + 1.94e11T^{2} \)
43 \( 1 + 8.96e5T + 2.71e11T^{2} \)
47 \( 1 + 2.13e4T + 5.06e11T^{2} \)
53 \( 1 + 9.10e5T + 1.17e12T^{2} \)
61 \( 1 - 8.10e5T + 3.14e12T^{2} \)
67 \( 1 + 3.83e6T + 6.06e12T^{2} \)
71 \( 1 - 3.35e6T + 9.09e12T^{2} \)
73 \( 1 - 4.61e5T + 1.10e13T^{2} \)
79 \( 1 + 1.37e6T + 1.92e13T^{2} \)
83 \( 1 - 7.63e6T + 2.71e13T^{2} \)
89 \( 1 - 5.26e6T + 4.42e13T^{2} \)
97 \( 1 + 9.23e6T + 8.07e13T^{2} \)
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

−9.490483095845369886472647006236, −8.793983856137915598411790387695, −8.184247629574211480451976180499, −6.96940533142793815790335690287, −6.11560554972336384846807075439, −4.88853173508552915352272885303, −3.86146692841586765904180655866, −3.44168606167820222781136446978, −1.62388659103032355425682398722, −0.28888081924939630822912019035, 0.28888081924939630822912019035, 1.62388659103032355425682398722, 3.44168606167820222781136446978, 3.86146692841586765904180655866, 4.88853173508552915352272885303, 6.11560554972336384846807075439, 6.96940533142793815790335690287, 8.184247629574211480451976180499, 8.793983856137915598411790387695, 9.490483095845369886472647006236

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