Properties

Degree 2
Conductor $ 2^{2} \cdot 19 \cdot 79 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.94·5-s − 0.969·7-s − 3·9-s + 0.202·11-s − 5.56·13-s + 3.01·17-s + 19-s + 6.09·23-s − 1.21·25-s − 6.46·29-s − 5.83·31-s + 1.88·35-s + 8.65·37-s − 7.53·41-s + 2.75·43-s + 5.83·45-s − 5.67·47-s − 6.06·49-s − 12.1·53-s − 0.393·55-s − 10.5·59-s + 1.88·61-s + 2.90·63-s + 10.8·65-s + 8.95·67-s + 5.27·71-s − 9.05·73-s + ⋯
L(s)  = 1  − 0.869·5-s − 0.366·7-s − 9-s + 0.0609·11-s − 1.54·13-s + 0.730·17-s + 0.229·19-s + 1.27·23-s − 0.243·25-s − 1.20·29-s − 1.04·31-s + 0.318·35-s + 1.42·37-s − 1.17·41-s + 0.420·43-s + 0.869·45-s − 0.828·47-s − 0.865·49-s − 1.67·53-s − 0.0530·55-s − 1.37·59-s + 0.241·61-s + 0.366·63-s + 1.34·65-s + 1.09·67-s + 0.626·71-s − 1.05·73-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6004\)    =    \(2^{2} \cdot 19 \cdot 79\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6004} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6835025011$
$L(\frac12)$  $\approx$  $0.6835025011$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;79\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;19,\;79\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
19 \( 1 - T \)
79 \( 1 - T \)
good3 \( 1 + 3T^{2} \)
5 \( 1 + 1.94T + 5T^{2} \)
7 \( 1 + 0.969T + 7T^{2} \)
11 \( 1 - 0.202T + 11T^{2} \)
13 \( 1 + 5.56T + 13T^{2} \)
17 \( 1 - 3.01T + 17T^{2} \)
23 \( 1 - 6.09T + 23T^{2} \)
29 \( 1 + 6.46T + 29T^{2} \)
31 \( 1 + 5.83T + 31T^{2} \)
37 \( 1 - 8.65T + 37T^{2} \)
41 \( 1 + 7.53T + 41T^{2} \)
43 \( 1 - 2.75T + 43T^{2} \)
47 \( 1 + 5.67T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 1.88T + 61T^{2} \)
67 \( 1 - 8.95T + 67T^{2} \)
71 \( 1 - 5.27T + 71T^{2} \)
73 \( 1 + 9.05T + 73T^{2} \)
83 \( 1 - 4.23T + 83T^{2} \)
89 \( 1 + 7.91T + 89T^{2} \)
97 \( 1 + 12.3T + 97T^{2} \)
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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

−7.84141604147809409879900609986, −7.57129409426107967510583673259, −6.78342029995345896432091876983, −5.87979304161633866089469136725, −5.17396708019918584802637470266, −4.52546608637103164519521465513, −3.38300390972777565606983060855, −3.06703124627373126351930895887, −1.92189871611579277591957116742, −0.41258655127817816240804506498, 0.41258655127817816240804506498, 1.92189871611579277591957116742, 3.06703124627373126351930895887, 3.38300390972777565606983060855, 4.52546608637103164519521465513, 5.17396708019918584802637470266, 5.87979304161633866089469136725, 6.78342029995345896432091876983, 7.57129409426107967510583673259, 7.84141604147809409879900609986

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