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Properties

Label 2-580-580.579-c0-0-5
Degree $2$
Conductor $580$
Sign $1$
Analytic cond. $0.289457$
Root an. cond. $0.538012$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 8-s + 9-s − 10-s + 16-s − 2·17-s + 18-s − 20-s + 25-s − 29-s + 32-s − 2·34-s + 36-s − 2·37-s − 40-s − 45-s − 49-s + 50-s − 58-s + 64-s − 2·68-s + 72-s + 2·73-s − 2·74-s − 80-s + ⋯
L(s)  = 1  + 2-s + 4-s − 5-s + 8-s + 9-s − 10-s + 16-s − 2·17-s + 18-s − 20-s + 25-s − 29-s + 32-s − 2·34-s + 36-s − 2·37-s − 40-s − 45-s − 49-s + 50-s − 58-s + 64-s − 2·68-s + 72-s + 2·73-s − 2·74-s − 80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(580\)    =    \(2^{2} \cdot 5 \cdot 29\)
Sign: $1$
Analytic conductor: \(0.289457\)
Root analytic conductor: \(0.538012\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{580} (579, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 580,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.462512404\)
\(L(\frac12)\) \(\approx\) \(1.462512404\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + T \)
29 \( 1 + T \)
good3 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 + T )^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( ( 1 + T )^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )^{2} \)
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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

−11.08736269062704702044037663996, −10.44269410440832008543693005495, −9.133384151229510254261521952647, −8.033810394382029849841899126232, −7.08976422143673490594864795564, −6.56753479260130438460504963730, −5.08273594341290074411568750469, −4.30167398370409115662946164621, −3.49430697897175681395203874488, −1.98364805092622174881950652951, 1.98364805092622174881950652951, 3.49430697897175681395203874488, 4.30167398370409115662946164621, 5.08273594341290074411568750469, 6.56753479260130438460504963730, 7.08976422143673490594864795564, 8.033810394382029849841899126232, 9.133384151229510254261521952647, 10.44269410440832008543693005495, 11.08736269062704702044037663996

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