Properties

Degree 2
Conductor 619
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 0.993·2-s + 2.38·3-s − 1.01·4-s + 2.26·5-s + 2.36·6-s + 3.26·7-s − 2.99·8-s + 2.67·9-s + 2.25·10-s − 3.48·11-s − 2.41·12-s − 0.0423·13-s + 3.24·14-s + 5.39·15-s − 0.950·16-s + 0.559·17-s + 2.65·18-s + 1.62·19-s − 2.29·20-s + 7.78·21-s − 3.46·22-s + 4.81·23-s − 7.12·24-s + 0.135·25-s − 0.0420·26-s − 0.782·27-s − 3.30·28-s + ⋯
L(s)  = 1  + 0.702·2-s + 1.37·3-s − 0.506·4-s + 1.01·5-s + 0.966·6-s + 1.23·7-s − 1.05·8-s + 0.890·9-s + 0.712·10-s − 1.05·11-s − 0.696·12-s − 0.0117·13-s + 0.868·14-s + 1.39·15-s − 0.237·16-s + 0.135·17-s + 0.625·18-s + 0.372·19-s − 0.513·20-s + 1.69·21-s − 0.739·22-s + 1.00·23-s − 1.45·24-s + 0.0271·25-s − 0.00825·26-s − 0.150·27-s − 0.625·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.27487$
$L(\frac12)$  $\approx$  $3.27487$
$L(\frac{3}{2})$   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 \( 1 - 0.993T + 2T^{2} \)
3 \( 1 - 2.38T + 3T^{2} \)
5 \( 1 - 2.26T + 5T^{2} \)
7 \( 1 - 3.26T + 7T^{2} \)
11 \( 1 + 3.48T + 11T^{2} \)
13 \( 1 + 0.0423T + 13T^{2} \)
17 \( 1 - 0.559T + 17T^{2} \)
19 \( 1 - 1.62T + 19T^{2} \)
23 \( 1 - 4.81T + 23T^{2} \)
29 \( 1 + 6.07T + 29T^{2} \)
31 \( 1 + 6.33T + 31T^{2} \)
37 \( 1 + 9.01T + 37T^{2} \)
41 \( 1 - 7.61T + 41T^{2} \)
43 \( 1 + 1.05T + 43T^{2} \)
47 \( 1 - 2.17T + 47T^{2} \)
53 \( 1 - 4.67T + 53T^{2} \)
59 \( 1 + 6.68T + 59T^{2} \)
61 \( 1 - 8.79T + 61T^{2} \)
67 \( 1 - 15.4T + 67T^{2} \)
71 \( 1 + 4.11T + 71T^{2} \)
73 \( 1 + 6.53T + 73T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 + 6.39T + 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 - 2.73T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.49272015005403510841542586022, −9.485822325164386492674855059798, −8.920390716680059181074395423256, −8.120630523886880474656257774553, −7.28848560305866619447433979508, −5.59563483712966710920440123207, −5.17265809849603903202916143413, −3.93403356461907855576023777188, −2.84607542187097599281255747957, −1.85277470008660528151292015325, 1.85277470008660528151292015325, 2.84607542187097599281255747957, 3.93403356461907855576023777188, 5.17265809849603903202916143413, 5.59563483712966710920440123207, 7.28848560305866619447433979508, 8.120630523886880474656257774553, 8.920390716680059181074395423256, 9.485822325164386492674855059798, 10.49272015005403510841542586022

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