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  + 2.81·2-s + 1.69·3-s + 5.94·4-s − 2.73·5-s + 4.77·6-s − 1.24·7-s + 11.1·8-s − 0.129·9-s − 7.69·10-s − 0.667·11-s + 10.0·12-s + 0.636·13-s − 3.51·14-s − 4.62·15-s + 19.4·16-s − 1.78·17-s − 0.363·18-s − 5.81·19-s − 16.2·20-s − 2.11·21-s − 1.88·22-s − 1.72·23-s + 18.8·24-s + 2.45·25-s + 1.79·26-s − 5.30·27-s − 7.40·28-s + ⋯
L(s)  = 1  + 1.99·2-s + 0.978·3-s + 2.97·4-s − 1.22·5-s + 1.94·6-s − 0.471·7-s + 3.92·8-s − 0.0430·9-s − 2.43·10-s − 0.201·11-s + 2.90·12-s + 0.176·13-s − 0.939·14-s − 1.19·15-s + 4.85·16-s − 0.433·17-s − 0.0856·18-s − 1.33·19-s − 3.62·20-s − 0.461·21-s − 0.400·22-s − 0.359·23-s + 3.84·24-s + 0.491·25-s + 0.351·26-s − 1.02·27-s − 1.40·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.16132$
$L(\frac12)$  $\approx$  $5.16132$
$L(\frac{3}{2})$   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 \( 1 - 2.81T + 2T^{2} \)
3 \( 1 - 1.69T + 3T^{2} \)
5 \( 1 + 2.73T + 5T^{2} \)
7 \( 1 + 1.24T + 7T^{2} \)
11 \( 1 + 0.667T + 11T^{2} \)
13 \( 1 - 0.636T + 13T^{2} \)
17 \( 1 + 1.78T + 17T^{2} \)
19 \( 1 + 5.81T + 19T^{2} \)
23 \( 1 + 1.72T + 23T^{2} \)
29 \( 1 - 5.11T + 29T^{2} \)
31 \( 1 - 2.93T + 31T^{2} \)
37 \( 1 + 2.26T + 37T^{2} \)
41 \( 1 - 12.5T + 41T^{2} \)
43 \( 1 + 0.542T + 43T^{2} \)
47 \( 1 + 7.59T + 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 - 5.40T + 59T^{2} \)
61 \( 1 - 7.79T + 61T^{2} \)
67 \( 1 - 10.9T + 67T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 + 2.62T + 79T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 + 0.970T + 89T^{2} \)
97 \( 1 + 6.09T + 97T^{2} \)
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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

−11.15459682297677273126072443450, −10.04817556198669582413392793804, −8.422700896106344999246726650314, −7.84317840437518248209202116957, −6.82594173650390036063897957689, −6.03634443375816023834731850774, −4.67599414915801600675596665872, −3.94794609656790578519764239538, −3.17931953570429837273854830600, −2.27544098013980643947967490297, 2.27544098013980643947967490297, 3.17931953570429837273854830600, 3.94794609656790578519764239538, 4.67599414915801600675596665872, 6.03634443375816023834731850774, 6.82594173650390036063897957689, 7.84317840437518248209202116957, 8.422700896106344999246726650314, 10.04817556198669582413392793804, 11.15459682297677273126072443450

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