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  + 2.07·2-s + 3.16·3-s + 2.32·4-s + 0.563·5-s + 6.57·6-s − 3.36·7-s + 0.676·8-s + 6.98·9-s + 1.17·10-s − 4.60·11-s + 7.34·12-s + 2.23·13-s − 7.00·14-s + 1.77·15-s − 3.24·16-s − 2.95·17-s + 14.5·18-s + 4.30·19-s + 1.30·20-s − 10.6·21-s − 9.58·22-s − 1.57·23-s + 2.13·24-s − 4.68·25-s + 4.64·26-s + 12.6·27-s − 7.82·28-s + ⋯
L(s)  = 1  + 1.47·2-s + 1.82·3-s + 1.16·4-s + 0.251·5-s + 2.68·6-s − 1.27·7-s + 0.239·8-s + 2.32·9-s + 0.370·10-s − 1.38·11-s + 2.12·12-s + 0.619·13-s − 1.87·14-s + 0.459·15-s − 0.810·16-s − 0.716·17-s + 3.42·18-s + 0.988·19-s + 0.292·20-s − 2.32·21-s − 2.04·22-s − 0.327·23-s + 0.436·24-s − 0.936·25-s + 0.910·26-s + 2.42·27-s − 1.47·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.80006$
$L(\frac12)$  $\approx$  $4.80006$
$L(\frac{3}{2})$   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 \( 1 - 2.07T + 2T^{2} \)
3 \( 1 - 3.16T + 3T^{2} \)
5 \( 1 - 0.563T + 5T^{2} \)
7 \( 1 + 3.36T + 7T^{2} \)
11 \( 1 + 4.60T + 11T^{2} \)
13 \( 1 - 2.23T + 13T^{2} \)
17 \( 1 + 2.95T + 17T^{2} \)
19 \( 1 - 4.30T + 19T^{2} \)
23 \( 1 + 1.57T + 23T^{2} \)
29 \( 1 - 9.78T + 29T^{2} \)
31 \( 1 + 0.992T + 31T^{2} \)
37 \( 1 - 1.16T + 37T^{2} \)
41 \( 1 - 4.00T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 - 2.33T + 47T^{2} \)
53 \( 1 - 3.76T + 53T^{2} \)
59 \( 1 - 5.06T + 59T^{2} \)
61 \( 1 + 3.80T + 61T^{2} \)
67 \( 1 + 5.68T + 67T^{2} \)
71 \( 1 + 5.17T + 71T^{2} \)
73 \( 1 - 10.8T + 73T^{2} \)
79 \( 1 - 5.82T + 79T^{2} \)
83 \( 1 + 0.352T + 83T^{2} \)
89 \( 1 + 7.55T + 89T^{2} \)
97 \( 1 - 17.4T + 97T^{2} \)
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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

−10.42328150626655749019797501407, −9.724596077578360325185781264183, −8.869453336039920675692319910794, −7.949300658793477025430291276793, −6.93616044707862727485443870828, −6.03484326379400860097251023928, −4.80396823284541140111984414233, −3.70777773041669629229407018547, −3.04017606575367137580434486422, −2.32019663115583922638974884521, 2.32019663115583922638974884521, 3.04017606575367137580434486422, 3.70777773041669629229407018547, 4.80396823284541140111984414233, 6.03484326379400860097251023928, 6.93616044707862727485443870828, 7.949300658793477025430291276793, 8.869453336039920675692319910794, 9.724596077578360325185781264183, 10.42328150626655749019797501407

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