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  − 0.938·2-s + 2.04·3-s − 1.11·4-s + 3.68·5-s − 1.92·6-s − 2.71·7-s + 2.92·8-s + 1.18·9-s − 3.46·10-s − 2.41·11-s − 2.28·12-s + 5.06·13-s + 2.55·14-s + 7.54·15-s − 0.509·16-s + 2.79·17-s − 1.11·18-s + 4.67·19-s − 4.12·20-s − 5.55·21-s + 2.26·22-s + 2.15·23-s + 5.98·24-s + 8.60·25-s − 4.75·26-s − 3.71·27-s + 3.04·28-s + ⋯
L(s)  = 1  − 0.663·2-s + 1.18·3-s − 0.559·4-s + 1.64·5-s − 0.783·6-s − 1.02·7-s + 1.03·8-s + 0.395·9-s − 1.09·10-s − 0.728·11-s − 0.660·12-s + 1.40·13-s + 0.681·14-s + 1.94·15-s − 0.127·16-s + 0.678·17-s − 0.262·18-s + 1.07·19-s − 0.922·20-s − 1.21·21-s + 0.483·22-s + 0.449·23-s + 1.22·24-s + 1.72·25-s − 0.932·26-s − 0.714·27-s + 0.574·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.65620$
$L(\frac12)$  $\approx$  $1.65620$
$L(\frac{3}{2})$   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 \( 1 + 0.938T + 2T^{2} \)
3 \( 1 - 2.04T + 3T^{2} \)
5 \( 1 - 3.68T + 5T^{2} \)
7 \( 1 + 2.71T + 7T^{2} \)
11 \( 1 + 2.41T + 11T^{2} \)
13 \( 1 - 5.06T + 13T^{2} \)
17 \( 1 - 2.79T + 17T^{2} \)
19 \( 1 - 4.67T + 19T^{2} \)
23 \( 1 - 2.15T + 23T^{2} \)
29 \( 1 - 0.405T + 29T^{2} \)
31 \( 1 + 2.87T + 31T^{2} \)
37 \( 1 - 4.37T + 37T^{2} \)
41 \( 1 + 2.91T + 41T^{2} \)
43 \( 1 - 4.39T + 43T^{2} \)
47 \( 1 + 4.68T + 47T^{2} \)
53 \( 1 - 5.90T + 53T^{2} \)
59 \( 1 - 2.92T + 59T^{2} \)
61 \( 1 + 8.81T + 61T^{2} \)
67 \( 1 - 6.63T + 67T^{2} \)
71 \( 1 - 1.75T + 71T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 + 17.0T + 79T^{2} \)
83 \( 1 + 9.17T + 83T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 + 6.48T + 97T^{2} \)
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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

−10.01704971179075392654616799481, −9.698605910601622948978197970639, −9.000945006061346155400715381398, −8.351056046801468537396618282440, −7.32884282358836983287553381388, −6.06780888894087029413180256663, −5.29375584344496105348722858488, −3.62389188238814567424070208609, −2.72072246623048306659908042891, −1.35060995591817130036840513814, 1.35060995591817130036840513814, 2.72072246623048306659908042891, 3.62389188238814567424070208609, 5.29375584344496105348722858488, 6.06780888894087029413180256663, 7.32884282358836983287553381388, 8.351056046801468537396618282440, 9.000945006061346155400715381398, 9.698605910601622948978197970639, 10.01704971179075392654616799481

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