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  + 2.38·2-s + 1.15·3-s + 3.70·4-s + 0.568·5-s + 2.75·6-s − 1.24·7-s + 4.07·8-s − 1.66·9-s + 1.35·10-s + 4.37·11-s + 4.28·12-s − 2.48·13-s − 2.96·14-s + 0.656·15-s + 2.33·16-s − 3.57·17-s − 3.98·18-s + 8.32·19-s + 2.10·20-s − 1.43·21-s + 10.4·22-s + 3.53·23-s + 4.71·24-s − 4.67·25-s − 5.94·26-s − 5.38·27-s − 4.60·28-s + ⋯
L(s)  = 1  + 1.68·2-s + 0.666·3-s + 1.85·4-s + 0.254·5-s + 1.12·6-s − 0.469·7-s + 1.44·8-s − 0.555·9-s + 0.429·10-s + 1.31·11-s + 1.23·12-s − 0.690·13-s − 0.792·14-s + 0.169·15-s + 0.582·16-s − 0.865·17-s − 0.938·18-s + 1.90·19-s + 0.471·20-s − 0.312·21-s + 2.22·22-s + 0.736·23-s + 0.961·24-s − 0.935·25-s − 1.16·26-s − 1.03·27-s − 0.870·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.42090$
$L(\frac12)$  $\approx$  $4.42090$
$L(\frac{3}{2})$   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 \( 1 - 2.38T + 2T^{2} \)
3 \( 1 - 1.15T + 3T^{2} \)
5 \( 1 - 0.568T + 5T^{2} \)
7 \( 1 + 1.24T + 7T^{2} \)
11 \( 1 - 4.37T + 11T^{2} \)
13 \( 1 + 2.48T + 13T^{2} \)
17 \( 1 + 3.57T + 17T^{2} \)
19 \( 1 - 8.32T + 19T^{2} \)
23 \( 1 - 3.53T + 23T^{2} \)
29 \( 1 + 3.16T + 29T^{2} \)
31 \( 1 + 8.22T + 31T^{2} \)
37 \( 1 - 3.99T + 37T^{2} \)
41 \( 1 + 8.44T + 41T^{2} \)
43 \( 1 - 3.77T + 43T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 + 8.87T + 53T^{2} \)
59 \( 1 + 13.5T + 59T^{2} \)
61 \( 1 - 7.69T + 61T^{2} \)
67 \( 1 - 8.17T + 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 - 6.99T + 73T^{2} \)
79 \( 1 - 8.19T + 79T^{2} \)
83 \( 1 + 16.0T + 83T^{2} \)
89 \( 1 - 17.3T + 89T^{2} \)
97 \( 1 + 1.96T + 97T^{2} \)
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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

−11.11140143102878145623075657840, −9.487986973127465657348688606520, −9.177798103588595837670558857718, −7.62781751456495716256281285730, −6.80555270006857317529626021492, −5.88480658708337725646191586477, −5.06011072440132353039632588450, −3.82362198128657371358000011657, −3.18530318389671275971120518570, −2.05673750614893676317374448759, 2.05673750614893676317374448759, 3.18530318389671275971120518570, 3.82362198128657371358000011657, 5.06011072440132353039632588450, 5.88480658708337725646191586477, 6.80555270006857317529626021492, 7.62781751456495716256281285730, 9.177798103588595837670558857718, 9.487986973127465657348688606520, 11.11140143102878145623075657840

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