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  − 1.76·5-s + 2.40·7-s − 3·9-s + 4.64·11-s + 3.69·13-s + 1.00·17-s + 19-s − 2.66·23-s − 1.88·25-s − 0.522·29-s + 7.54·31-s − 4.24·35-s − 4.36·37-s + 9.51·41-s − 0.856·43-s + 5.29·45-s + 9.01·47-s − 1.22·49-s − 12.1·53-s − 8.19·55-s + 12.6·59-s − 2.58·61-s − 7.20·63-s − 6.53·65-s + 6.77·67-s − 2.48·71-s − 5.73·73-s + ⋯
L(s)  = 1  − 0.789·5-s + 0.908·7-s − 9-s + 1.39·11-s + 1.02·13-s + 0.244·17-s + 0.229·19-s − 0.556·23-s − 0.376·25-s − 0.0970·29-s + 1.35·31-s − 0.717·35-s − 0.718·37-s + 1.48·41-s − 0.130·43-s + 0.789·45-s + 1.31·47-s − 0.175·49-s − 1.67·53-s − 1.10·55-s + 1.65·59-s − 0.331·61-s − 0.908·63-s − 0.810·65-s + 0.827·67-s − 0.295·71-s − 0.671·73-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  =  $(2,\ 6004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.013349092$
$L(\frac12)$  $\approx$  $2.013349092$
$L(\frac{3}{2})$   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 \( 1 + 1.76T + 5T^{2} \)
7 \( 1 - 2.40T + 7T^{2} \)
11 \( 1 - 4.64T + 11T^{2} \)
13 \( 1 - 3.69T + 13T^{2} \)
17 \( 1 - 1.00T + 17T^{2} \)
23 \( 1 + 2.66T + 23T^{2} \)
29 \( 1 + 0.522T + 29T^{2} \)
31 \( 1 - 7.54T + 31T^{2} \)
37 \( 1 + 4.36T + 37T^{2} \)
41 \( 1 - 9.51T + 41T^{2} \)
43 \( 1 + 0.856T + 43T^{2} \)
47 \( 1 - 9.01T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 - 12.6T + 59T^{2} \)
61 \( 1 + 2.58T + 61T^{2} \)
67 \( 1 - 6.77T + 67T^{2} \)
71 \( 1 + 2.48T + 71T^{2} \)
73 \( 1 + 5.73T + 73T^{2} \)
83 \( 1 - 5.22T + 83T^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 + 4.55T + 97T^{2} \)
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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

−8.113385239934177939313633015365, −7.57918324014519034377528712404, −6.56371078893826405370555243789, −6.01785058799246263467425604222, −5.22540548258154970001720586413, −4.22133528727801328248160596674, −3.82624245465945797212437259714, −2.89363710043659577501309702926, −1.71708694913121798316435329284, −0.78437141148040842709414059503, 0.78437141148040842709414059503, 1.71708694913121798316435329284, 2.89363710043659577501309702926, 3.82624245465945797212437259714, 4.22133528727801328248160596674, 5.22540548258154970001720586413, 6.01785058799246263467425604222, 6.56371078893826405370555243789, 7.57918324014519034377528712404, 8.113385239934177939313633015365

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