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  + 1.77·2-s − 2.61·3-s + 1.14·4-s + 2.86·5-s − 4.64·6-s + 2.04·7-s − 1.52·8-s + 3.85·9-s + 5.07·10-s − 3.53·11-s − 2.99·12-s + 5.12·13-s + 3.63·14-s − 7.49·15-s − 4.97·16-s + 5.62·17-s + 6.83·18-s + 4.67·19-s + 3.26·20-s − 5.36·21-s − 6.26·22-s − 1.06·23-s + 3.98·24-s + 3.19·25-s + 9.08·26-s − 2.24·27-s + 2.34·28-s + ⋯
L(s)  = 1  + 1.25·2-s − 1.51·3-s + 0.570·4-s + 1.28·5-s − 1.89·6-s + 0.774·7-s − 0.537·8-s + 1.28·9-s + 1.60·10-s − 1.06·11-s − 0.863·12-s + 1.42·13-s + 0.971·14-s − 1.93·15-s − 1.24·16-s + 1.36·17-s + 1.61·18-s + 1.07·19-s + 0.731·20-s − 1.17·21-s − 1.33·22-s − 0.221·23-s + 0.812·24-s + 0.639·25-s + 1.78·26-s − 0.432·27-s + 0.442·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.18342$
$L(\frac12)$  $\approx$  $2.18342$
$L(\frac{3}{2})$   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 \( 1 - 1.77T + 2T^{2} \)
3 \( 1 + 2.61T + 3T^{2} \)
5 \( 1 - 2.86T + 5T^{2} \)
7 \( 1 - 2.04T + 7T^{2} \)
11 \( 1 + 3.53T + 11T^{2} \)
13 \( 1 - 5.12T + 13T^{2} \)
17 \( 1 - 5.62T + 17T^{2} \)
19 \( 1 - 4.67T + 19T^{2} \)
23 \( 1 + 1.06T + 23T^{2} \)
29 \( 1 - 8.34T + 29T^{2} \)
31 \( 1 + 4.96T + 31T^{2} \)
37 \( 1 + 1.19T + 37T^{2} \)
41 \( 1 + 1.05T + 41T^{2} \)
43 \( 1 - 6.18T + 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + 2.73T + 53T^{2} \)
59 \( 1 + 10.0T + 59T^{2} \)
61 \( 1 + 6.59T + 61T^{2} \)
67 \( 1 + 14.7T + 67T^{2} \)
71 \( 1 - 8.64T + 71T^{2} \)
73 \( 1 + 9.61T + 73T^{2} \)
79 \( 1 - 2.15T + 79T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 + 3.00T + 89T^{2} \)
97 \( 1 - 2.92T + 97T^{2} \)
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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

−10.76704717240101632079015379938, −10.20036581766084799150301775478, −9.025900766687343736359704296677, −7.69236383858799453328088604510, −6.37706266533298875489777900286, −5.60937980820098280442345897087, −5.47458109263617739811987622505, −4.47901046129733400605351991646, −3.01209693063152303368128706889, −1.32197642990595936093293754730, 1.32197642990595936093293754730, 3.01209693063152303368128706889, 4.47901046129733400605351991646, 5.47458109263617739811987622505, 5.60937980820098280442345897087, 6.37706266533298875489777900286, 7.69236383858799453328088604510, 9.025900766687343736359704296677, 10.20036581766084799150301775478, 10.76704717240101632079015379938

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