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  + 2.58·2-s − 2.01·3-s + 4.66·4-s − 1.70·5-s − 5.20·6-s + 3.16·7-s + 6.86·8-s + 1.06·9-s − 4.39·10-s + 3.07·11-s − 9.39·12-s + 1.76·13-s + 8.17·14-s + 3.43·15-s + 8.40·16-s + 3.12·17-s + 2.74·18-s − 1.60·19-s − 7.93·20-s − 6.38·21-s + 7.94·22-s + 5.10·23-s − 13.8·24-s − 2.09·25-s + 4.55·26-s + 3.90·27-s + 14.7·28-s + ⋯
L(s)  = 1  + 1.82·2-s − 1.16·3-s + 2.33·4-s − 0.761·5-s − 2.12·6-s + 1.19·7-s + 2.42·8-s + 0.354·9-s − 1.38·10-s + 0.928·11-s − 2.71·12-s + 0.490·13-s + 2.18·14-s + 0.886·15-s + 2.10·16-s + 0.759·17-s + 0.646·18-s − 0.368·19-s − 1.77·20-s − 1.39·21-s + 1.69·22-s + 1.06·23-s − 2.82·24-s − 0.419·25-s + 0.894·26-s + 0.751·27-s + 2.78·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.12629$
$L(\frac12)$  $\approx$  $3.12629$
$L(\frac{3}{2})$   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 \( 1 - 2.58T + 2T^{2} \)
3 \( 1 + 2.01T + 3T^{2} \)
5 \( 1 + 1.70T + 5T^{2} \)
7 \( 1 - 3.16T + 7T^{2} \)
11 \( 1 - 3.07T + 11T^{2} \)
13 \( 1 - 1.76T + 13T^{2} \)
17 \( 1 - 3.12T + 17T^{2} \)
19 \( 1 + 1.60T + 19T^{2} \)
23 \( 1 - 5.10T + 23T^{2} \)
29 \( 1 + 0.870T + 29T^{2} \)
31 \( 1 + 5.41T + 31T^{2} \)
37 \( 1 - 1.53T + 37T^{2} \)
41 \( 1 + 9.86T + 41T^{2} \)
43 \( 1 + 1.06T + 43T^{2} \)
47 \( 1 + 6.84T + 47T^{2} \)
53 \( 1 - 8.24T + 53T^{2} \)
59 \( 1 - 3.71T + 59T^{2} \)
61 \( 1 + 6.66T + 61T^{2} \)
67 \( 1 - 4.94T + 67T^{2} \)
71 \( 1 + 15.3T + 71T^{2} \)
73 \( 1 + 9.17T + 73T^{2} \)
79 \( 1 - 0.733T + 79T^{2} \)
83 \( 1 + 1.05T + 83T^{2} \)
89 \( 1 - 1.86T + 89T^{2} \)
97 \( 1 - 5.89T + 97T^{2} \)
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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

−11.31536159029055966160869319695, −10.42711059362301608803066148874, −8.644048067561710261842472305926, −7.50957185837064491956929430674, −6.67450395256576958814342278977, −5.76362220088502454430104143751, −5.05394281529176604085802362135, −4.27896800501851115406870182472, −3.35343066335089801056962522926, −1.53979467988529118867202299012, 1.53979467988529118867202299012, 3.35343066335089801056962522926, 4.27896800501851115406870182472, 5.05394281529176604085802362135, 5.76362220088502454430104143751, 6.67450395256576958814342278977, 7.50957185837064491956929430674, 8.644048067561710261842472305926, 10.42711059362301608803066148874, 11.31536159029055966160869319695

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