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  − 2.00·3-s − 2.46·5-s + 4.06·7-s + 1.00·9-s − 3.25·11-s + 4.74·13-s + 4.93·15-s − 4.01·17-s + 19-s − 8.14·21-s − 2.85·23-s + 1.07·25-s + 3.98·27-s − 3.94·29-s + 2.73·31-s + 6.50·33-s − 10.0·35-s + 2.47·37-s − 9.50·39-s + 7.55·41-s + 2.10·43-s − 2.48·45-s − 12.7·47-s + 9.53·49-s + 8.03·51-s + 13.6·53-s + 8.01·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.10·5-s + 1.53·7-s + 0.336·9-s − 0.980·11-s + 1.31·13-s + 1.27·15-s − 0.973·17-s + 0.229·19-s − 1.77·21-s − 0.596·23-s + 0.214·25-s + 0.767·27-s − 0.731·29-s + 0.491·31-s + 1.13·33-s − 1.69·35-s + 0.406·37-s − 1.52·39-s + 1.17·41-s + 0.321·43-s − 0.370·45-s − 1.85·47-s + 1.36·49-s + 1.12·51-s + 1.87·53-s + 1.08·55-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 \( 1 + 2.00T + 3T^{2} \)
5 \( 1 + 2.46T + 5T^{2} \)
7 \( 1 - 4.06T + 7T^{2} \)
11 \( 1 + 3.25T + 11T^{2} \)
13 \( 1 - 4.74T + 13T^{2} \)
17 \( 1 + 4.01T + 17T^{2} \)
23 \( 1 + 2.85T + 23T^{2} \)
29 \( 1 + 3.94T + 29T^{2} \)
31 \( 1 - 2.73T + 31T^{2} \)
37 \( 1 - 2.47T + 37T^{2} \)
41 \( 1 - 7.55T + 41T^{2} \)
43 \( 1 - 2.10T + 43T^{2} \)
47 \( 1 + 12.7T + 47T^{2} \)
53 \( 1 - 13.6T + 53T^{2} \)
59 \( 1 - 6.69T + 59T^{2} \)
61 \( 1 + 14.4T + 61T^{2} \)
67 \( 1 - 2.77T + 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 + 9.87T + 73T^{2} \)
83 \( 1 + 13.5T + 83T^{2} \)
89 \( 1 + 1.17T + 89T^{2} \)
97 \( 1 - 7.33e - 5T + 97T^{2} \)
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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

−7.67587737353842410458465607834, −7.18212351453145351040777473355, −6.01176648772706416259023530179, −5.71277308471914527957644598632, −4.60678834068399516143413002902, −4.48003983246692912011520151682, −3.39450913559179770710691433326, −2.17231974017791806845221768486, −1.07134899847730775915191384810, 0, 1.07134899847730775915191384810, 2.17231974017791806845221768486, 3.39450913559179770710691433326, 4.48003983246692912011520151682, 4.60678834068399516143413002902, 5.71277308471914527957644598632, 6.01176648772706416259023530179, 7.18212351453145351040777473355, 7.67587737353842410458465607834

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