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  − 0.394·2-s + 1.27·3-s − 1.84·4-s + 1.92·5-s − 0.502·6-s + 3.31·7-s + 1.51·8-s − 1.37·9-s − 0.760·10-s + 2.52·11-s − 2.34·12-s − 4.15·13-s − 1.30·14-s + 2.45·15-s + 3.09·16-s + 2.64·17-s + 0.543·18-s + 6.63·19-s − 3.55·20-s + 4.22·21-s − 0.996·22-s − 4.34·23-s + 1.93·24-s − 1.28·25-s + 1.64·26-s − 5.57·27-s − 6.11·28-s + ⋯
L(s)  = 1  − 0.278·2-s + 0.735·3-s − 0.922·4-s + 0.862·5-s − 0.205·6-s + 1.25·7-s + 0.536·8-s − 0.459·9-s − 0.240·10-s + 0.761·11-s − 0.677·12-s − 1.15·13-s − 0.349·14-s + 0.634·15-s + 0.772·16-s + 0.640·17-s + 0.128·18-s + 1.52·19-s − 0.795·20-s + 0.921·21-s − 0.212·22-s − 0.905·23-s + 0.394·24-s − 0.256·25-s + 0.321·26-s − 1.07·27-s − 1.15·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.73036$
$L(\frac12)$  $\approx$  $1.73036$
$L(\frac{3}{2})$   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 \( 1 + 0.394T + 2T^{2} \)
3 \( 1 - 1.27T + 3T^{2} \)
5 \( 1 - 1.92T + 5T^{2} \)
7 \( 1 - 3.31T + 7T^{2} \)
11 \( 1 - 2.52T + 11T^{2} \)
13 \( 1 + 4.15T + 13T^{2} \)
17 \( 1 - 2.64T + 17T^{2} \)
19 \( 1 - 6.63T + 19T^{2} \)
23 \( 1 + 4.34T + 23T^{2} \)
29 \( 1 - 0.725T + 29T^{2} \)
31 \( 1 - 6.55T + 31T^{2} \)
37 \( 1 - 6.31T + 37T^{2} \)
41 \( 1 + 0.0915T + 41T^{2} \)
43 \( 1 - 0.725T + 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 + 4.81T + 53T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 + 3.71T + 61T^{2} \)
67 \( 1 + 7.85T + 67T^{2} \)
71 \( 1 + 8.16T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 + 2.55T + 83T^{2} \)
89 \( 1 + 9.14T + 89T^{2} \)
97 \( 1 - 0.730T + 97T^{2} \)
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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

−10.19847089956165860178441700948, −9.625490723393769058257237639125, −8.985020101052622270627111803999, −8.032317973588889519425251752808, −7.54791709402811663542247085130, −5.85678353986373043863107064432, −5.09592096654851356374539169410, −4.06676241454453968017860622484, −2.64436977122787549211386920970, −1.34785198323514141736890921103, 1.34785198323514141736890921103, 2.64436977122787549211386920970, 4.06676241454453968017860622484, 5.09592096654851356374539169410, 5.85678353986373043863107064432, 7.54791709402811663542247085130, 8.032317973588889519425251752808, 8.985020101052622270627111803999, 9.625490723393769058257237639125, 10.19847089956165860178441700948

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