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  + 0.262·2-s − 0.800·3-s − 1.93·4-s − 2.40·5-s − 0.209·6-s + 1.00·7-s − 1.03·8-s − 2.35·9-s − 0.631·10-s + 2.12·11-s + 1.54·12-s − 0.383·13-s + 0.264·14-s + 1.92·15-s + 3.59·16-s + 3.95·17-s − 0.618·18-s + 3.69·19-s + 4.65·20-s − 0.807·21-s + 0.555·22-s + 4.85·23-s + 0.824·24-s + 0.802·25-s − 0.100·26-s + 4.28·27-s − 1.94·28-s + ⋯
L(s)  = 1  + 0.185·2-s − 0.461·3-s − 0.965·4-s − 1.07·5-s − 0.0856·6-s + 0.381·7-s − 0.364·8-s − 0.786·9-s − 0.199·10-s + 0.639·11-s + 0.446·12-s − 0.106·13-s + 0.0706·14-s + 0.497·15-s + 0.898·16-s + 0.959·17-s − 0.145·18-s + 0.848·19-s + 1.04·20-s − 0.176·21-s + 0.118·22-s + 1.01·23-s + 0.168·24-s + 0.160·25-s − 0.0197·26-s + 0.825·27-s − 0.368·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.835464$
$L(\frac12)$  $\approx$  $0.835464$
$L(\frac{3}{2})$   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 \( 1 - 0.262T + 2T^{2} \)
3 \( 1 + 0.800T + 3T^{2} \)
5 \( 1 + 2.40T + 5T^{2} \)
7 \( 1 - 1.00T + 7T^{2} \)
11 \( 1 - 2.12T + 11T^{2} \)
13 \( 1 + 0.383T + 13T^{2} \)
17 \( 1 - 3.95T + 17T^{2} \)
19 \( 1 - 3.69T + 19T^{2} \)
23 \( 1 - 4.85T + 23T^{2} \)
29 \( 1 + 3.08T + 29T^{2} \)
31 \( 1 - 0.275T + 31T^{2} \)
37 \( 1 + 3.06T + 37T^{2} \)
41 \( 1 - 2.32T + 41T^{2} \)
43 \( 1 - 8.80T + 43T^{2} \)
47 \( 1 - 1.90T + 47T^{2} \)
53 \( 1 - 8.51T + 53T^{2} \)
59 \( 1 - 0.748T + 59T^{2} \)
61 \( 1 - 8.47T + 61T^{2} \)
67 \( 1 + 7.23T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + 3.23T + 73T^{2} \)
79 \( 1 + 5.96T + 79T^{2} \)
83 \( 1 - 2.05T + 83T^{2} \)
89 \( 1 + 6.75T + 89T^{2} \)
97 \( 1 - 1.61T + 97T^{2} \)
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

−10.81614740186200507545928119388, −9.645727291457475230866271763874, −8.828130690471232143400906658336, −8.039686255328189643070798256213, −7.20049225542117878473079438031, −5.79436110283242235410854804714, −5.09529225722850486678025381287, −4.05709891338098635631056534721, −3.18905624376828693971482150638, −0.806724464850011905546358673447, 0.806724464850011905546358673447, 3.18905624376828693971482150638, 4.05709891338098635631056534721, 5.09529225722850486678025381287, 5.79436110283242235410854804714, 7.20049225542117878473079438031, 8.039686255328189643070798256213, 8.828130690471232143400906658336, 9.645727291457475230866271763874, 10.81614740186200507545928119388

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