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  + 0.271·2-s − 1.01·3-s − 1.92·4-s + 4.11·5-s − 0.276·6-s + 2.70·7-s − 1.06·8-s − 1.96·9-s + 1.11·10-s + 2.02·11-s + 1.96·12-s + 1.50·13-s + 0.736·14-s − 4.19·15-s + 3.56·16-s − 3.18·17-s − 0.533·18-s − 6.45·19-s − 7.93·20-s − 2.75·21-s + 0.550·22-s + 8.15·23-s + 1.08·24-s + 11.9·25-s + 0.408·26-s + 5.05·27-s − 5.21·28-s + ⋯
L(s)  = 1  + 0.192·2-s − 0.587·3-s − 0.963·4-s + 1.84·5-s − 0.112·6-s + 1.02·7-s − 0.377·8-s − 0.654·9-s + 0.354·10-s + 0.611·11-s + 0.565·12-s + 0.416·13-s + 0.196·14-s − 1.08·15-s + 0.890·16-s − 0.772·17-s − 0.125·18-s − 1.48·19-s − 1.77·20-s − 0.601·21-s + 0.117·22-s + 1.70·23-s + 0.221·24-s + 2.39·25-s + 0.0800·26-s + 0.972·27-s − 0.986·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.59696$
$L(\frac12)$  $\approx$  $1.59696$
$L(\frac{3}{2})$   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 \( 1 - 0.271T + 2T^{2} \)
3 \( 1 + 1.01T + 3T^{2} \)
5 \( 1 - 4.11T + 5T^{2} \)
7 \( 1 - 2.70T + 7T^{2} \)
11 \( 1 - 2.02T + 11T^{2} \)
13 \( 1 - 1.50T + 13T^{2} \)
17 \( 1 + 3.18T + 17T^{2} \)
19 \( 1 + 6.45T + 19T^{2} \)
23 \( 1 - 8.15T + 23T^{2} \)
29 \( 1 - 2.94T + 29T^{2} \)
31 \( 1 + 0.223T + 31T^{2} \)
37 \( 1 - 11.8T + 37T^{2} \)
41 \( 1 + 0.427T + 41T^{2} \)
43 \( 1 - 2.47T + 43T^{2} \)
47 \( 1 + 4.43T + 47T^{2} \)
53 \( 1 - 1.65T + 53T^{2} \)
59 \( 1 + 8.46T + 59T^{2} \)
61 \( 1 - 5.39T + 61T^{2} \)
67 \( 1 + 5.24T + 67T^{2} \)
71 \( 1 + 3.81T + 71T^{2} \)
73 \( 1 - 13.1T + 73T^{2} \)
79 \( 1 - 3.12T + 79T^{2} \)
83 \( 1 + 17.6T + 83T^{2} \)
89 \( 1 - 4.40T + 89T^{2} \)
97 \( 1 + 8.58T + 97T^{2} \)
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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

−10.77686417498565245118130121982, −9.649924185242280551670206714357, −8.918069491471507616199210760822, −8.381837399833864685056135841890, −6.59123288867519418213676628743, −5.98892762058509875800544983118, −5.11637006761272242291192523007, −4.45048628573988412554217375393, −2.63927170890810956394160981036, −1.24246628498784892666841127290, 1.24246628498784892666841127290, 2.63927170890810956394160981036, 4.45048628573988412554217375393, 5.11637006761272242291192523007, 5.98892762058509875800544983118, 6.59123288867519418213676628743, 8.381837399833864685056135841890, 8.918069491471507616199210760822, 9.649924185242280551670206714357, 10.77686417498565245118130121982

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