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  + 1.39·2-s + 2.10·3-s − 0.0457·4-s + 3.90·5-s + 2.94·6-s − 1.82·7-s − 2.85·8-s + 1.43·9-s + 5.46·10-s + 4.06·11-s − 0.0963·12-s − 2.56·13-s − 2.54·14-s + 8.22·15-s − 3.90·16-s + 2.96·17-s + 2.00·18-s − 3.33·19-s − 0.178·20-s − 3.83·21-s + 5.68·22-s − 9.09·23-s − 6.02·24-s + 10.2·25-s − 3.58·26-s − 3.30·27-s + 0.0834·28-s + ⋯
L(s)  = 1  + 0.988·2-s + 1.21·3-s − 0.0228·4-s + 1.74·5-s + 1.20·6-s − 0.688·7-s − 1.01·8-s + 0.477·9-s + 1.72·10-s + 1.22·11-s − 0.0278·12-s − 0.712·13-s − 0.680·14-s + 2.12·15-s − 0.976·16-s + 0.717·17-s + 0.471·18-s − 0.764·19-s − 0.0400·20-s − 0.837·21-s + 1.21·22-s − 1.89·23-s − 1.22·24-s + 2.05·25-s − 0.703·26-s − 0.635·27-s + 0.0157·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.67966$
$L(\frac12)$  $\approx$  $3.67966$
$L(\frac{3}{2})$   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 \( 1 - 1.39T + 2T^{2} \)
3 \( 1 - 2.10T + 3T^{2} \)
5 \( 1 - 3.90T + 5T^{2} \)
7 \( 1 + 1.82T + 7T^{2} \)
11 \( 1 - 4.06T + 11T^{2} \)
13 \( 1 + 2.56T + 13T^{2} \)
17 \( 1 - 2.96T + 17T^{2} \)
19 \( 1 + 3.33T + 19T^{2} \)
23 \( 1 + 9.09T + 23T^{2} \)
29 \( 1 - 7.13T + 29T^{2} \)
31 \( 1 - 3.95T + 31T^{2} \)
37 \( 1 + 2.67T + 37T^{2} \)
41 \( 1 + 9.73T + 41T^{2} \)
43 \( 1 - 3.74T + 43T^{2} \)
47 \( 1 + 6.42T + 47T^{2} \)
53 \( 1 - 1.59T + 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 + 1.47T + 67T^{2} \)
71 \( 1 - 15.1T + 71T^{2} \)
73 \( 1 + 7.51T + 73T^{2} \)
79 \( 1 - 1.32T + 79T^{2} \)
83 \( 1 - 9.23T + 83T^{2} \)
89 \( 1 - 6.11T + 89T^{2} \)
97 \( 1 - 14.0T + 97T^{2} \)
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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

−10.09480778275744326765761596793, −9.759803290381886199502616543777, −9.030477530516247313921340948734, −8.239489304088623233535091435336, −6.55102802903778658139380057832, −6.16975324151510698743937464973, −5.03630715887532020112064597445, −3.85051250312354486238210345980, −2.91317086754811013460864324720, −1.96485365607532540361497522307, 1.96485365607532540361497522307, 2.91317086754811013460864324720, 3.85051250312354486238210345980, 5.03630715887532020112064597445, 6.16975324151510698743937464973, 6.55102802903778658139380057832, 8.239489304088623233535091435336, 9.030477530516247313921340948734, 9.759803290381886199502616543777, 10.09480778275744326765761596793

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