Properties

Degree 2
Conductor 619
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.45·2-s − 3.26·3-s + 0.117·4-s − 2.69·5-s − 4.74·6-s − 1.44·7-s − 2.73·8-s + 7.65·9-s − 3.92·10-s + 4.50·11-s − 0.382·12-s + 0.198·13-s − 2.09·14-s + 8.81·15-s − 4.22·16-s − 1.93·17-s + 11.1·18-s + 3.86·19-s − 0.316·20-s + 4.70·21-s + 6.55·22-s + 0.808·23-s + 8.94·24-s + 2.28·25-s + 0.288·26-s − 15.1·27-s − 0.168·28-s + ⋯
L(s)  = 1  + 1.02·2-s − 1.88·3-s + 0.0585·4-s − 1.20·5-s − 1.93·6-s − 0.545·7-s − 0.968·8-s + 2.55·9-s − 1.24·10-s + 1.35·11-s − 0.110·12-s + 0.0549·13-s − 0.560·14-s + 2.27·15-s − 1.05·16-s − 0.469·17-s + 2.62·18-s + 0.886·19-s − 0.0706·20-s + 1.02·21-s + 1.39·22-s + 0.168·23-s + 1.82·24-s + 0.457·25-s + 0.0565·26-s − 2.92·27-s − 0.0319·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(619\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{619} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.807776$
$L(\frac12)$  $\approx$  $0.807776$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 619$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 619$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad619 \( 1 - T \)
good2 \( 1 - 1.45T + 2T^{2} \)
3 \( 1 + 3.26T + 3T^{2} \)
5 \( 1 + 2.69T + 5T^{2} \)
7 \( 1 + 1.44T + 7T^{2} \)
11 \( 1 - 4.50T + 11T^{2} \)
13 \( 1 - 0.198T + 13T^{2} \)
17 \( 1 + 1.93T + 17T^{2} \)
19 \( 1 - 3.86T + 19T^{2} \)
23 \( 1 - 0.808T + 23T^{2} \)
29 \( 1 - 6.96T + 29T^{2} \)
31 \( 1 - 1.02T + 31T^{2} \)
37 \( 1 - 6.22T + 37T^{2} \)
41 \( 1 - 5.92T + 41T^{2} \)
43 \( 1 - 1.11T + 43T^{2} \)
47 \( 1 + 7.77T + 47T^{2} \)
53 \( 1 - 5.75T + 53T^{2} \)
59 \( 1 + 7.95T + 59T^{2} \)
61 \( 1 - 6.64T + 61T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 + 7.35T + 71T^{2} \)
73 \( 1 - 8.20T + 73T^{2} \)
79 \( 1 - 7.47T + 79T^{2} \)
83 \( 1 - 15.5T + 83T^{2} \)
89 \( 1 - 16.4T + 89T^{2} \)
97 \( 1 + 18.6T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.16575391132970090538559314611, −9.965934365531276838188880432658, −9.051952168826873695556534338535, −7.56196712844711328467298824689, −6.51582127372908591405087714304, −6.13983290624885724973987628270, −4.90090115096382194463216562252, −4.32083754122322093353627247144, −3.46260861271092601150247264389, −0.73865660663455951245009933203, 0.73865660663455951245009933203, 3.46260861271092601150247264389, 4.32083754122322093353627247144, 4.90090115096382194463216562252, 6.13983290624885724973987628270, 6.51582127372908591405087714304, 7.56196712844711328467298824689, 9.051952168826873695556534338535, 9.965934365531276838188880432658, 11.16575391132970090538559314611

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