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  − 1.37·2-s − 0.725·3-s − 0.121·4-s − 0.225·5-s + 0.994·6-s + 2.47·7-s + 2.90·8-s − 2.47·9-s + 0.308·10-s − 4.06·11-s + 0.0882·12-s + 2.26·13-s − 3.38·14-s + 0.163·15-s − 3.74·16-s + 7.19·17-s + 3.38·18-s − 7.47·19-s + 0.0273·20-s − 1.79·21-s + 5.56·22-s − 0.178·23-s − 2.11·24-s − 4.94·25-s − 3.10·26-s + 3.97·27-s − 0.300·28-s + ⋯
L(s)  = 1  − 0.969·2-s − 0.419·3-s − 0.0607·4-s − 0.100·5-s + 0.406·6-s + 0.934·7-s + 1.02·8-s − 0.824·9-s + 0.0975·10-s − 1.22·11-s + 0.0254·12-s + 0.628·13-s − 0.905·14-s + 0.0421·15-s − 0.935·16-s + 1.74·17-s + 0.798·18-s − 1.71·19-s + 0.00611·20-s − 0.391·21-s + 1.18·22-s − 0.0371·23-s − 0.430·24-s − 0.989·25-s − 0.608·26-s + 0.764·27-s − 0.0568·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.645822$
$L(\frac12)$  $\approx$  $0.645822$
$L(\frac{3}{2})$   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 \( 1 + 1.37T + 2T^{2} \)
3 \( 1 + 0.725T + 3T^{2} \)
5 \( 1 + 0.225T + 5T^{2} \)
7 \( 1 - 2.47T + 7T^{2} \)
11 \( 1 + 4.06T + 11T^{2} \)
13 \( 1 - 2.26T + 13T^{2} \)
17 \( 1 - 7.19T + 17T^{2} \)
19 \( 1 + 7.47T + 19T^{2} \)
23 \( 1 + 0.178T + 23T^{2} \)
29 \( 1 - 10.5T + 29T^{2} \)
31 \( 1 - 1.27T + 31T^{2} \)
37 \( 1 - 2.08T + 37T^{2} \)
41 \( 1 - 4.23T + 41T^{2} \)
43 \( 1 - 8.52T + 43T^{2} \)
47 \( 1 - 9.96T + 47T^{2} \)
53 \( 1 - 0.619T + 53T^{2} \)
59 \( 1 - 11.8T + 59T^{2} \)
61 \( 1 - 6.30T + 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + 9.50T + 71T^{2} \)
73 \( 1 - 5.14T + 73T^{2} \)
79 \( 1 + 16.3T + 79T^{2} \)
83 \( 1 - 4.79T + 83T^{2} \)
89 \( 1 + 1.64T + 89T^{2} \)
97 \( 1 - 3.34T + 97T^{2} \)
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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

−10.56472351021641540393461806373, −9.904341327369648925018167917695, −8.487098956041875842499890100594, −8.321160047047178620052345431506, −7.47712218711359426710170030675, −6.02303477723435028528571373735, −5.19785926794801550532565994363, −4.16414165644554453334526805049, −2.45443668544270497767264085633, −0.834929733689204520348011313208, 0.834929733689204520348011313208, 2.45443668544270497767264085633, 4.16414165644554453334526805049, 5.19785926794801550532565994363, 6.02303477723435028528571373735, 7.47712218711359426710170030675, 8.321160047047178620052345431506, 8.487098956041875842499890100594, 9.904341327369648925018167917695, 10.56472351021641540393461806373

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