Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.68·3-s − 3.32·5-s + 3.49·7-s − 0.173·9-s + 1.26·11-s − 1.98·13-s + 5.58·15-s − 1.03·17-s + 19-s − 5.87·21-s − 6.45·23-s + 6.02·25-s + 5.33·27-s + 5.24·29-s − 0.427·31-s − 2.12·33-s − 11.6·35-s + 8.20·37-s + 3.33·39-s − 12.4·41-s − 4.70·43-s + 0.575·45-s + 9.04·47-s + 5.22·49-s + 1.73·51-s + 0.909·53-s − 4.20·55-s + ⋯
L(s)  = 1  − 0.970·3-s − 1.48·5-s + 1.32·7-s − 0.0578·9-s + 0.381·11-s − 0.550·13-s + 1.44·15-s − 0.250·17-s + 0.229·19-s − 1.28·21-s − 1.34·23-s + 1.20·25-s + 1.02·27-s + 0.973·29-s − 0.0767·31-s − 0.370·33-s − 1.96·35-s + 1.34·37-s + 0.533·39-s − 1.94·41-s − 0.718·43-s + 0.0858·45-s + 1.31·47-s + 0.746·49-s + 0.243·51-s + 0.124·53-s − 0.567·55-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  =  1
Selberg data  =  $(2,\ 6004,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 1.68T + 3T^{2} \)
5 \( 1 + 3.32T + 5T^{2} \)
7 \( 1 - 3.49T + 7T^{2} \)
11 \( 1 - 1.26T + 11T^{2} \)
13 \( 1 + 1.98T + 13T^{2} \)
17 \( 1 + 1.03T + 17T^{2} \)
23 \( 1 + 6.45T + 23T^{2} \)
29 \( 1 - 5.24T + 29T^{2} \)
31 \( 1 + 0.427T + 31T^{2} \)
37 \( 1 - 8.20T + 37T^{2} \)
41 \( 1 + 12.4T + 41T^{2} \)
43 \( 1 + 4.70T + 43T^{2} \)
47 \( 1 - 9.04T + 47T^{2} \)
53 \( 1 - 0.909T + 53T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 15.1T + 61T^{2} \)
67 \( 1 + 3.76T + 67T^{2} \)
71 \( 1 - 5.33T + 71T^{2} \)
73 \( 1 - 5.75T + 73T^{2} \)
83 \( 1 + 8.11T + 83T^{2} \)
89 \( 1 + 2.12T + 89T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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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

−7.82983088752623947215568720642, −7.05400282950216529393786307227, −6.33748927272201247138596384912, −5.44284788719919250006058799980, −4.74668885854992284571992451840, −4.31011357489466134542689210043, −3.42265691973845571234475851177, −2.24389218972198047985699598431, −1.02664138539665128640691545062, 0, 1.02664138539665128640691545062, 2.24389218972198047985699598431, 3.42265691973845571234475851177, 4.31011357489466134542689210043, 4.74668885854992284571992451840, 5.44284788719919250006058799980, 6.33748927272201247138596384912, 7.05400282950216529393786307227, 7.82983088752623947215568720642

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