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  − 1.67·2-s − 2.69·3-s + 0.797·4-s + 3.55·5-s + 4.51·6-s + 2.20·7-s + 2.01·8-s + 4.27·9-s − 5.95·10-s + 1.00·11-s − 2.14·12-s + 1.66·13-s − 3.67·14-s − 9.60·15-s − 4.95·16-s + 0.528·17-s − 7.15·18-s + 2.00·19-s + 2.83·20-s − 5.93·21-s − 1.68·22-s − 6.13·23-s − 5.42·24-s + 7.66·25-s − 2.78·26-s − 3.44·27-s + 1.75·28-s + ⋯
L(s)  = 1  − 1.18·2-s − 1.55·3-s + 0.398·4-s + 1.59·5-s + 1.84·6-s + 0.831·7-s + 0.711·8-s + 1.42·9-s − 1.88·10-s + 0.304·11-s − 0.620·12-s + 0.462·13-s − 0.983·14-s − 2.47·15-s − 1.23·16-s + 0.128·17-s − 1.68·18-s + 0.459·19-s + 0.634·20-s − 1.29·21-s − 0.359·22-s − 1.27·23-s − 1.10·24-s + 1.53·25-s − 0.547·26-s − 0.662·27-s + 0.331·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.696156$
$L(\frac12)$  $\approx$  $0.696156$
$L(\frac{3}{2})$   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 \( 1 + 1.67T + 2T^{2} \)
3 \( 1 + 2.69T + 3T^{2} \)
5 \( 1 - 3.55T + 5T^{2} \)
7 \( 1 - 2.20T + 7T^{2} \)
11 \( 1 - 1.00T + 11T^{2} \)
13 \( 1 - 1.66T + 13T^{2} \)
17 \( 1 - 0.528T + 17T^{2} \)
19 \( 1 - 2.00T + 19T^{2} \)
23 \( 1 + 6.13T + 23T^{2} \)
29 \( 1 + 3.47T + 29T^{2} \)
31 \( 1 - 7.58T + 31T^{2} \)
37 \( 1 + 0.759T + 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 + 0.631T + 47T^{2} \)
53 \( 1 - 12.7T + 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 4.33T + 61T^{2} \)
67 \( 1 + 9.29T + 67T^{2} \)
71 \( 1 + 3.23T + 71T^{2} \)
73 \( 1 - 0.177T + 73T^{2} \)
79 \( 1 + 9.62T + 79T^{2} \)
83 \( 1 - 2.48T + 83T^{2} \)
89 \( 1 - 3.02T + 89T^{2} \)
97 \( 1 - 3.43T + 97T^{2} \)
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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

−10.23733735614073096482069611083, −10.10552876391718477223582767591, −9.073815153153239551266987267201, −8.124668631043700365865054664222, −6.97032286162618961626369113680, −6.06990912142542526544129653103, −5.40241747421711009265417127419, −4.43652682503560190571093398531, −1.95526282336867629364559990759, −1.01496464563590593437914778616, 1.01496464563590593437914778616, 1.95526282336867629364559990759, 4.43652682503560190571093398531, 5.40241747421711009265417127419, 6.06990912142542526544129653103, 6.97032286162618961626369113680, 8.124668631043700365865054664222, 9.073815153153239551266987267201, 10.10552876391718477223582767591, 10.23733735614073096482069611083

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