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  + 1.54·2-s + 0.456·3-s + 0.402·4-s + 0.324·5-s + 0.707·6-s + 2.73·7-s − 2.47·8-s − 2.79·9-s + 0.502·10-s + 5.58·11-s + 0.183·12-s + 6.14·13-s + 4.24·14-s + 0.147·15-s − 4.64·16-s − 1.80·17-s − 4.32·18-s + 1.89·19-s + 0.130·20-s + 1.24·21-s + 8.65·22-s + 0.651·23-s − 1.13·24-s − 4.89·25-s + 9.53·26-s − 2.64·27-s + 1.10·28-s + ⋯
L(s)  = 1  + 1.09·2-s + 0.263·3-s + 0.201·4-s + 0.144·5-s + 0.288·6-s + 1.03·7-s − 0.875·8-s − 0.930·9-s + 0.158·10-s + 1.68·11-s + 0.0530·12-s + 1.70·13-s + 1.13·14-s + 0.0382·15-s − 1.16·16-s − 0.437·17-s − 1.01·18-s + 0.435·19-s + 0.0291·20-s + 0.272·21-s + 1.84·22-s + 0.135·23-s − 0.230·24-s − 0.978·25-s + 1.86·26-s − 0.508·27-s + 0.208·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.89467$
$L(\frac12)$  $\approx$  $2.89467$
$L(\frac{3}{2})$   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 \( 1 - 1.54T + 2T^{2} \)
3 \( 1 - 0.456T + 3T^{2} \)
5 \( 1 - 0.324T + 5T^{2} \)
7 \( 1 - 2.73T + 7T^{2} \)
11 \( 1 - 5.58T + 11T^{2} \)
13 \( 1 - 6.14T + 13T^{2} \)
17 \( 1 + 1.80T + 17T^{2} \)
19 \( 1 - 1.89T + 19T^{2} \)
23 \( 1 - 0.651T + 23T^{2} \)
29 \( 1 - 3.25T + 29T^{2} \)
31 \( 1 - 7.40T + 31T^{2} \)
37 \( 1 + 11.3T + 37T^{2} \)
41 \( 1 - 4.27T + 41T^{2} \)
43 \( 1 + 8.46T + 43T^{2} \)
47 \( 1 + 6.21T + 47T^{2} \)
53 \( 1 + 8.37T + 53T^{2} \)
59 \( 1 - 8.13T + 59T^{2} \)
61 \( 1 + 15.0T + 61T^{2} \)
67 \( 1 - 0.0636T + 67T^{2} \)
71 \( 1 - 5.20T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 + 0.958T + 83T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 + 4.18T + 97T^{2} \)
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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

−11.09798487317286711209584345696, −9.598520926101837601765599195107, −8.667180458813392802991198341913, −8.304921836189942097797184912026, −6.60820308590946192113980955395, −6.01571767052915364928022900975, −4.97908343385799248747526347199, −4.00365536488285460152890386671, −3.21450652158866582355242683996, −1.56667067300482226653643235754, 1.56667067300482226653643235754, 3.21450652158866582355242683996, 4.00365536488285460152890386671, 4.97908343385799248747526347199, 6.01571767052915364928022900975, 6.60820308590946192113980955395, 8.304921836189942097797184912026, 8.667180458813392802991198341913, 9.598520926101837601765599195107, 11.09798487317286711209584345696

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