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  + 2.70·2-s − 2.72·3-s + 5.33·4-s + 4.10·5-s − 7.38·6-s − 3.88·7-s + 9.04·8-s + 4.42·9-s + 11.1·10-s + 4.15·11-s − 14.5·12-s − 2.28·13-s − 10.5·14-s − 11.1·15-s + 13.8·16-s − 2.82·17-s + 11.9·18-s − 1.09·19-s + 21.9·20-s + 10.5·21-s + 11.2·22-s − 2.23·23-s − 24.6·24-s + 11.8·25-s − 6.19·26-s − 3.88·27-s − 20.7·28-s + ⋯
L(s)  = 1  + 1.91·2-s − 1.57·3-s + 2.66·4-s + 1.83·5-s − 3.01·6-s − 1.46·7-s + 3.19·8-s + 1.47·9-s + 3.51·10-s + 1.25·11-s − 4.19·12-s − 0.634·13-s − 2.81·14-s − 2.88·15-s + 3.45·16-s − 0.684·17-s + 2.82·18-s − 0.250·19-s + 4.89·20-s + 2.30·21-s + 2.39·22-s − 0.465·23-s − 5.02·24-s + 2.36·25-s − 1.21·26-s − 0.747·27-s − 3.91·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.51665$
$L(\frac12)$  $\approx$  $3.51665$
$L(\frac{3}{2})$   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 \( 1 - 2.70T + 2T^{2} \)
3 \( 1 + 2.72T + 3T^{2} \)
5 \( 1 - 4.10T + 5T^{2} \)
7 \( 1 + 3.88T + 7T^{2} \)
11 \( 1 - 4.15T + 11T^{2} \)
13 \( 1 + 2.28T + 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 + 1.09T + 19T^{2} \)
23 \( 1 + 2.23T + 23T^{2} \)
29 \( 1 - 5.30T + 29T^{2} \)
31 \( 1 + 2.19T + 31T^{2} \)
37 \( 1 + 2.22T + 37T^{2} \)
41 \( 1 - 4.99T + 41T^{2} \)
43 \( 1 + 6.91T + 43T^{2} \)
47 \( 1 + 7.22T + 47T^{2} \)
53 \( 1 + 10.2T + 53T^{2} \)
59 \( 1 - 2.59T + 59T^{2} \)
61 \( 1 + 14.7T + 61T^{2} \)
67 \( 1 + 4.92T + 67T^{2} \)
71 \( 1 + 2.08T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 - 7.96T + 79T^{2} \)
83 \( 1 + 6.16T + 83T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 - 4.60T + 97T^{2} \)
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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

−10.84120427908856178614313921256, −10.12264071550444594516637046645, −9.397065896261994899595840201897, −6.86481078068071681122147455093, −6.43792149344802571871688012789, −6.12331000944940042134560261738, −5.24953205839294673135605246519, −4.41099633450007289866735916360, −3.02021993946710082609492465360, −1.72326303544247175209811883673, 1.72326303544247175209811883673, 3.02021993946710082609492465360, 4.41099633450007289866735916360, 5.24953205839294673135605246519, 6.12331000944940042134560261738, 6.43792149344802571871688012789, 6.86481078068071681122147455093, 9.397065896261994899595840201897, 10.12264071550444594516637046645, 10.84120427908856178614313921256

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