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  + 1.95·2-s + 2.74·3-s + 1.83·4-s − 1.58·5-s + 5.37·6-s + 2.21·7-s − 0.314·8-s + 4.51·9-s − 3.11·10-s + 2.06·11-s + 5.04·12-s − 2.72·13-s + 4.33·14-s − 4.35·15-s − 4.29·16-s − 0.241·17-s + 8.84·18-s − 6.25·19-s − 2.92·20-s + 6.05·21-s + 4.04·22-s + 2.33·23-s − 0.862·24-s − 2.47·25-s − 5.34·26-s + 4.15·27-s + 4.06·28-s + ⋯
L(s)  = 1  + 1.38·2-s + 1.58·3-s + 0.919·4-s − 0.710·5-s + 2.19·6-s + 0.835·7-s − 0.111·8-s + 1.50·9-s − 0.983·10-s + 0.622·11-s + 1.45·12-s − 0.756·13-s + 1.15·14-s − 1.12·15-s − 1.07·16-s − 0.0585·17-s + 2.08·18-s − 1.43·19-s − 0.652·20-s + 1.32·21-s + 0.862·22-s + 0.485·23-s − 0.176·24-s − 0.495·25-s − 1.04·26-s + 0.798·27-s + 0.768·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.34269$
$L(\frac12)$  $\approx$  $4.34269$
$L(\frac{3}{2})$   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 \( 1 - 1.95T + 2T^{2} \)
3 \( 1 - 2.74T + 3T^{2} \)
5 \( 1 + 1.58T + 5T^{2} \)
7 \( 1 - 2.21T + 7T^{2} \)
11 \( 1 - 2.06T + 11T^{2} \)
13 \( 1 + 2.72T + 13T^{2} \)
17 \( 1 + 0.241T + 17T^{2} \)
19 \( 1 + 6.25T + 19T^{2} \)
23 \( 1 - 2.33T + 23T^{2} \)
29 \( 1 + 5.85T + 29T^{2} \)
31 \( 1 - 6.65T + 31T^{2} \)
37 \( 1 - 8.08T + 37T^{2} \)
41 \( 1 - 0.133T + 41T^{2} \)
43 \( 1 - 9.11T + 43T^{2} \)
47 \( 1 - 5.04T + 47T^{2} \)
53 \( 1 - 6.24T + 53T^{2} \)
59 \( 1 + 3.77T + 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 + 7.79T + 67T^{2} \)
71 \( 1 + 2.92T + 71T^{2} \)
73 \( 1 + 4.05T + 73T^{2} \)
79 \( 1 + 2.05T + 79T^{2} \)
83 \( 1 - 11.2T + 83T^{2} \)
89 \( 1 + 14.6T + 89T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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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

−10.88482184041753140626476031995, −9.503426716360453057064634085885, −8.771076543074297533649408125847, −7.924578160990398038445936178068, −7.18312124266119686760464692793, −5.94444184467805293912524856346, −4.41297513414760794836974610731, −4.22994799975670815368761785989, −3.03623515692918011504967528314, −2.09774347593157156926293277581, 2.09774347593157156926293277581, 3.03623515692918011504967528314, 4.22994799975670815368761785989, 4.41297513414760794836974610731, 5.94444184467805293912524856346, 7.18312124266119686760464692793, 7.924578160990398038445936178068, 8.771076543074297533649408125847, 9.503426716360453057064634085885, 10.88482184041753140626476031995

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