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  + 2.46·2-s − 0.157·3-s + 4.08·4-s + 2.37·5-s − 0.389·6-s + 4.70·7-s + 5.13·8-s − 2.97·9-s + 5.86·10-s − 3.89·11-s − 0.644·12-s − 6.62·13-s + 11.6·14-s − 0.375·15-s + 4.49·16-s − 5.95·17-s − 7.33·18-s − 0.395·19-s + 9.70·20-s − 0.743·21-s − 9.60·22-s − 1.84·23-s − 0.810·24-s + 0.653·25-s − 16.3·26-s + 0.943·27-s + 19.2·28-s + ⋯
L(s)  = 1  + 1.74·2-s − 0.0911·3-s + 2.04·4-s + 1.06·5-s − 0.158·6-s + 1.77·7-s + 1.81·8-s − 0.991·9-s + 1.85·10-s − 1.17·11-s − 0.185·12-s − 1.83·13-s + 3.10·14-s − 0.0969·15-s + 1.12·16-s − 1.44·17-s − 1.72·18-s − 0.0906·19-s + 2.16·20-s − 0.162·21-s − 2.04·22-s − 0.385·23-s − 0.165·24-s + 0.130·25-s − 3.20·26-s + 0.181·27-s + 3.63·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.35448$
$L(\frac12)$  $\approx$  $4.35448$
$L(\frac{3}{2})$   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 \( 1 - 2.46T + 2T^{2} \)
3 \( 1 + 0.157T + 3T^{2} \)
5 \( 1 - 2.37T + 5T^{2} \)
7 \( 1 - 4.70T + 7T^{2} \)
11 \( 1 + 3.89T + 11T^{2} \)
13 \( 1 + 6.62T + 13T^{2} \)
17 \( 1 + 5.95T + 17T^{2} \)
19 \( 1 + 0.395T + 19T^{2} \)
23 \( 1 + 1.84T + 23T^{2} \)
29 \( 1 - 10.6T + 29T^{2} \)
31 \( 1 - 7.56T + 31T^{2} \)
37 \( 1 + 5.85T + 37T^{2} \)
41 \( 1 - 1.13T + 41T^{2} \)
43 \( 1 - 3.18T + 43T^{2} \)
47 \( 1 - 0.625T + 47T^{2} \)
53 \( 1 - 12.8T + 53T^{2} \)
59 \( 1 + 0.141T + 59T^{2} \)
61 \( 1 + 3.91T + 61T^{2} \)
67 \( 1 - 0.202T + 67T^{2} \)
71 \( 1 - 6.99T + 71T^{2} \)
73 \( 1 - 0.952T + 73T^{2} \)
79 \( 1 + 8.10T + 79T^{2} \)
83 \( 1 - 9.46T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−10.83785138095375222940113470499, −10.21024098216866125196838700550, −8.697635968721919110704798311913, −7.77124314844195959435707590156, −6.67870697437598678853682679275, −5.65633045469255006571933422940, −4.97169719647278394634446627885, −4.57741022369577636987548766878, −2.48295386106412492564925226448, −2.33060413800315170043219839925, 2.33060413800315170043219839925, 2.48295386106412492564925226448, 4.57741022369577636987548766878, 4.97169719647278394634446627885, 5.65633045469255006571933422940, 6.67870697437598678853682679275, 7.77124314844195959435707590156, 8.697635968721919110704798311913, 10.21024098216866125196838700550, 10.83785138095375222940113470499

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