Properties

Label 2-1328-83.82-c0-0-2
Degree $2$
Conductor $1328$
Sign $1$
Analytic cond. $0.662758$
Root an. cond. $0.814099$
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  + 3-s + 7-s + 11-s − 17-s + 21-s − 2·23-s + 25-s − 27-s − 29-s + 31-s + 33-s − 37-s + 2·41-s − 51-s + 59-s − 61-s − 2·69-s + 75-s + 77-s − 81-s − 83-s − 87-s + 93-s − 109-s − 111-s − 113-s − 119-s + ⋯
L(s)  = 1  + 3-s + 7-s + 11-s − 17-s + 21-s − 2·23-s + 25-s − 27-s − 29-s + 31-s + 33-s − 37-s + 2·41-s − 51-s + 59-s − 61-s − 2·69-s + 75-s + 77-s − 81-s − 83-s − 87-s + 93-s − 109-s − 111-s − 113-s − 119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1328\)    =    \(2^{4} \cdot 83\)
Sign: $1$
Analytic conductor: \(0.662758\)
Root analytic conductor: \(0.814099\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1328} (497, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1328,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.585022164\)
\(L(\frac12)\) \(\approx\) \(1.585022164\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.559465075735668886731535413435, −8.918781378133019943236457369013, −8.287123648012125383867424831268, −7.63886962710502613850075455603, −6.61793116824543450857127683638, −5.68312089006156878812846942551, −4.46864943192264282147295107706, −3.82511842162550708535569567788, −2.56236355163019656337205767838, −1.67036991833096906207723721274, 1.67036991833096906207723721274, 2.56236355163019656337205767838, 3.82511842162550708535569567788, 4.46864943192264282147295107706, 5.68312089006156878812846942551, 6.61793116824543450857127683638, 7.63886962710502613850075455603, 8.287123648012125383867424831268, 8.918781378133019943236457369013, 9.559465075735668886731535413435

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