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  − 1.73·2-s − 0.518·3-s + 1.01·4-s − 1.50·5-s + 0.899·6-s − 4.51·7-s + 1.71·8-s − 2.73·9-s + 2.60·10-s − 3.30·11-s − 0.523·12-s + 1.38·13-s + 7.82·14-s + 0.779·15-s − 4.99·16-s − 0.194·17-s + 4.73·18-s + 2.90·19-s − 1.51·20-s + 2.33·21-s + 5.73·22-s + 2.25·23-s − 0.890·24-s − 2.74·25-s − 2.40·26-s + 2.97·27-s − 4.55·28-s + ⋯
L(s)  = 1  − 1.22·2-s − 0.299·3-s + 0.505·4-s − 0.672·5-s + 0.367·6-s − 1.70·7-s + 0.607·8-s − 0.910·9-s + 0.824·10-s − 0.996·11-s − 0.151·12-s + 0.384·13-s + 2.09·14-s + 0.201·15-s − 1.24·16-s − 0.0470·17-s + 1.11·18-s + 0.667·19-s − 0.339·20-s + 0.510·21-s + 1.22·22-s + 0.469·23-s − 0.181·24-s − 0.548·25-s − 0.471·26-s + 0.571·27-s − 0.861·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.240490$
$L(\frac12)$  $\approx$  $0.240490$
$L(\frac{3}{2})$   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 \( 1 + 1.73T + 2T^{2} \)
3 \( 1 + 0.518T + 3T^{2} \)
5 \( 1 + 1.50T + 5T^{2} \)
7 \( 1 + 4.51T + 7T^{2} \)
11 \( 1 + 3.30T + 11T^{2} \)
13 \( 1 - 1.38T + 13T^{2} \)
17 \( 1 + 0.194T + 17T^{2} \)
19 \( 1 - 2.90T + 19T^{2} \)
23 \( 1 - 2.25T + 23T^{2} \)
29 \( 1 + 1.21T + 29T^{2} \)
31 \( 1 + 2.87T + 31T^{2} \)
37 \( 1 - 2.81T + 37T^{2} \)
41 \( 1 - 8.65T + 41T^{2} \)
43 \( 1 + 4.18T + 43T^{2} \)
47 \( 1 + 4.41T + 47T^{2} \)
53 \( 1 - 6.95T + 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 + 4.50T + 61T^{2} \)
67 \( 1 + 14.1T + 67T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 + 1.49T + 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 + 8.73T + 83T^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 - 5.37T + 97T^{2} \)
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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

−10.48870925559356448667690865189, −9.642032194633381120389849474799, −8.999621695429499581422543577999, −8.067513589531143689058713635547, −7.32934062225382796319553653656, −6.32358861249574110559685516574, −5.29615126049886867750763883725, −3.77513312829909698013449211463, −2.69585187862172137786369299252, −0.48848415051343897176240104551, 0.48848415051343897176240104551, 2.69585187862172137786369299252, 3.77513312829909698013449211463, 5.29615126049886867750763883725, 6.32358861249574110559685516574, 7.32934062225382796319553653656, 8.067513589531143689058713635547, 8.999621695429499581422543577999, 9.642032194633381120389849474799, 10.48870925559356448667690865189

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