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  − 0.386·2-s − 1.55·3-s − 1.85·4-s − 3.75·5-s + 0.600·6-s − 2.84·7-s + 1.48·8-s − 0.587·9-s + 1.45·10-s + 0.341·11-s + 2.87·12-s − 4.58·13-s + 1.10·14-s + 5.82·15-s + 3.12·16-s − 4.50·17-s + 0.227·18-s − 4.60·19-s + 6.94·20-s + 4.42·21-s − 0.131·22-s − 6.46·23-s − 2.31·24-s + 9.06·25-s + 1.77·26-s + 5.57·27-s + 5.26·28-s + ⋯
L(s)  = 1  − 0.273·2-s − 0.896·3-s − 0.925·4-s − 1.67·5-s + 0.245·6-s − 1.07·7-s + 0.526·8-s − 0.195·9-s + 0.458·10-s + 0.102·11-s + 0.829·12-s − 1.27·13-s + 0.294·14-s + 1.50·15-s + 0.781·16-s − 1.09·17-s + 0.0535·18-s − 1.05·19-s + 1.55·20-s + 0.964·21-s − 0.0281·22-s − 1.34·23-s − 0.472·24-s + 1.81·25-s + 0.347·26-s + 1.07·27-s + 0.995·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.0636388$
$L(\frac12)$  $\approx$  $0.0636388$
$L(\frac{3}{2})$   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 \( 1 + 0.386T + 2T^{2} \)
3 \( 1 + 1.55T + 3T^{2} \)
5 \( 1 + 3.75T + 5T^{2} \)
7 \( 1 + 2.84T + 7T^{2} \)
11 \( 1 - 0.341T + 11T^{2} \)
13 \( 1 + 4.58T + 13T^{2} \)
17 \( 1 + 4.50T + 17T^{2} \)
19 \( 1 + 4.60T + 19T^{2} \)
23 \( 1 + 6.46T + 23T^{2} \)
29 \( 1 - 9.68T + 29T^{2} \)
31 \( 1 - 7.65T + 31T^{2} \)
37 \( 1 + 0.361T + 37T^{2} \)
41 \( 1 + 7.84T + 41T^{2} \)
43 \( 1 + 9.06T + 43T^{2} \)
47 \( 1 - 2.68T + 47T^{2} \)
53 \( 1 + 1.93T + 53T^{2} \)
59 \( 1 + 0.518T + 59T^{2} \)
61 \( 1 + 7.90T + 61T^{2} \)
67 \( 1 - 6.35T + 67T^{2} \)
71 \( 1 + 6.34T + 71T^{2} \)
73 \( 1 + 13.6T + 73T^{2} \)
79 \( 1 + 12.1T + 79T^{2} \)
83 \( 1 + 5.17T + 83T^{2} \)
89 \( 1 - 0.605T + 89T^{2} \)
97 \( 1 + 0.592T + 97T^{2} \)
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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

−10.47533059154947439281212610698, −9.964225609694205849680056934217, −8.655561108195490985124446576465, −8.212684810923393971481405377217, −7.00198377498718291331980533926, −6.24724590260545764618968855583, −4.75716598725790353372355829722, −4.32093092285993047116871322779, −3.03342590860552507585939976467, −0.22359698785891014393310962308, 0.22359698785891014393310962308, 3.03342590860552507585939976467, 4.32093092285993047116871322779, 4.75716598725790353372355829722, 6.24724590260545764618968855583, 7.00198377498718291331980533926, 8.212684810923393971481405377217, 8.655561108195490985124446576465, 9.964225609694205849680056934217, 10.47533059154947439281212610698

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