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  − 2.61·2-s + 0.252·3-s + 4.85·4-s + 3.30·5-s − 0.660·6-s − 1.70·7-s − 7.47·8-s − 2.93·9-s − 8.65·10-s + 5.06·11-s + 1.22·12-s + 5.40·13-s + 4.47·14-s + 0.833·15-s + 9.86·16-s − 0.784·17-s + 7.68·18-s + 1.17·19-s + 16.0·20-s − 0.430·21-s − 13.2·22-s − 1.79·23-s − 1.88·24-s + 5.92·25-s − 14.1·26-s − 1.49·27-s − 8.30·28-s + ⋯
L(s)  = 1  − 1.85·2-s + 0.145·3-s + 2.42·4-s + 1.47·5-s − 0.269·6-s − 0.646·7-s − 2.64·8-s − 0.978·9-s − 2.73·10-s + 1.52·11-s + 0.353·12-s + 1.49·13-s + 1.19·14-s + 0.215·15-s + 2.46·16-s − 0.190·17-s + 1.81·18-s + 0.269·19-s + 3.58·20-s − 0.0940·21-s − 2.82·22-s − 0.374·23-s − 0.384·24-s + 1.18·25-s − 2.77·26-s − 0.288·27-s − 1.56·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.878057$
$L(\frac12)$  $\approx$  $0.878057$
$L(\frac{3}{2})$   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 \( 1 + 2.61T + 2T^{2} \)
3 \( 1 - 0.252T + 3T^{2} \)
5 \( 1 - 3.30T + 5T^{2} \)
7 \( 1 + 1.70T + 7T^{2} \)
11 \( 1 - 5.06T + 11T^{2} \)
13 \( 1 - 5.40T + 13T^{2} \)
17 \( 1 + 0.784T + 17T^{2} \)
19 \( 1 - 1.17T + 19T^{2} \)
23 \( 1 + 1.79T + 23T^{2} \)
29 \( 1 - 7.10T + 29T^{2} \)
31 \( 1 + 9.66T + 31T^{2} \)
37 \( 1 + 3.34T + 37T^{2} \)
41 \( 1 - 8.63T + 41T^{2} \)
43 \( 1 - 1.57T + 43T^{2} \)
47 \( 1 - 4.16T + 47T^{2} \)
53 \( 1 + 1.42T + 53T^{2} \)
59 \( 1 - 9.64T + 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 + 0.285T + 71T^{2} \)
73 \( 1 + 16.4T + 73T^{2} \)
79 \( 1 - 7.56T + 79T^{2} \)
83 \( 1 + 2.48T + 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + 0.0677T + 97T^{2} \)
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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

−10.33252087163809259982240458122, −9.511147867875494730166005147942, −8.966901905977217627194341335098, −8.532378731569592699421309085384, −7.11122608790616271868252040268, −6.19945568928341931744043206651, −5.91320127558740713138822347535, −3.46746296393431818013397926608, −2.22007345682876857145305559323, −1.14079133280932028941658238897, 1.14079133280932028941658238897, 2.22007345682876857145305559323, 3.46746296393431818013397926608, 5.91320127558740713138822347535, 6.19945568928341931744043206651, 7.11122608790616271868252040268, 8.532378731569592699421309085384, 8.966901905977217627194341335098, 9.511147867875494730166005147942, 10.33252087163809259982240458122

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