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  + 2.34·2-s + 0.501·3-s + 3.50·4-s + 2.53·5-s + 1.17·6-s − 0.903·7-s + 3.53·8-s − 2.74·9-s + 5.95·10-s − 1.96·11-s + 1.76·12-s + 3.76·13-s − 2.12·14-s + 1.27·15-s + 1.29·16-s + 4.17·17-s − 6.44·18-s − 3.60·19-s + 8.89·20-s − 0.453·21-s − 4.62·22-s − 0.374·23-s + 1.77·24-s + 1.42·25-s + 8.83·26-s − 2.88·27-s − 3.17·28-s + ⋯
L(s)  = 1  + 1.65·2-s + 0.289·3-s + 1.75·4-s + 1.13·5-s + 0.480·6-s − 0.341·7-s + 1.25·8-s − 0.916·9-s + 1.88·10-s − 0.593·11-s + 0.508·12-s + 1.04·13-s − 0.566·14-s + 0.328·15-s + 0.322·16-s + 1.01·17-s − 1.52·18-s − 0.826·19-s + 1.98·20-s − 0.0990·21-s − 0.985·22-s − 0.0781·23-s + 0.362·24-s + 0.285·25-s + 1.73·26-s − 0.555·27-s − 0.599·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.25631$
$L(\frac12)$  $\approx$  $4.25631$
$L(\frac{3}{2})$   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 \( 1 - 2.34T + 2T^{2} \)
3 \( 1 - 0.501T + 3T^{2} \)
5 \( 1 - 2.53T + 5T^{2} \)
7 \( 1 + 0.903T + 7T^{2} \)
11 \( 1 + 1.96T + 11T^{2} \)
13 \( 1 - 3.76T + 13T^{2} \)
17 \( 1 - 4.17T + 17T^{2} \)
19 \( 1 + 3.60T + 19T^{2} \)
23 \( 1 + 0.374T + 23T^{2} \)
29 \( 1 + 7.23T + 29T^{2} \)
31 \( 1 + 0.918T + 31T^{2} \)
37 \( 1 - 6.03T + 37T^{2} \)
41 \( 1 - 4.61T + 41T^{2} \)
43 \( 1 + 2.12T + 43T^{2} \)
47 \( 1 + 4.26T + 47T^{2} \)
53 \( 1 - 5.29T + 53T^{2} \)
59 \( 1 - 8.26T + 59T^{2} \)
61 \( 1 - 5.99T + 61T^{2} \)
67 \( 1 + 1.02T + 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 - 3.45T + 73T^{2} \)
79 \( 1 + 1.04T + 79T^{2} \)
83 \( 1 - 12.6T + 83T^{2} \)
89 \( 1 + 0.804T + 89T^{2} \)
97 \( 1 - 1.25T + 97T^{2} \)
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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

−10.89762921368632536836651605049, −9.883451304809993540862273209700, −8.918535585326032995775629117286, −7.82158688269244044757088596299, −6.48308868492573180295957835399, −5.82594768169620701154601726972, −5.33442396452248341040607924111, −3.92524286248829784312358522282, −3.01347562557292393802384484967, −2.04574174974082331927372056624, 2.04574174974082331927372056624, 3.01347562557292393802384484967, 3.92524286248829784312358522282, 5.33442396452248341040607924111, 5.82594768169620701154601726972, 6.48308868492573180295957835399, 7.82158688269244044757088596299, 8.918535585326032995775629117286, 9.883451304809993540862273209700, 10.89762921368632536836651605049

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