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  − 2.81·3-s − 0.243·5-s + 0.929·7-s + 4.93·9-s + 4.38·11-s − 1.17·13-s + 0.685·15-s + 6.41·17-s + 19-s − 2.61·21-s − 6.17·23-s − 4.94·25-s − 5.46·27-s − 5.16·29-s − 8.29·31-s − 12.3·33-s − 0.226·35-s − 11.1·37-s + 3.29·39-s + 3.11·41-s + 5.53·43-s − 1.20·45-s + 8.81·47-s − 6.13·49-s − 18.0·51-s + 10.2·53-s − 1.06·55-s + ⋯
L(s)  = 1  − 1.62·3-s − 0.108·5-s + 0.351·7-s + 1.64·9-s + 1.32·11-s − 0.324·13-s + 0.177·15-s + 1.55·17-s + 0.229·19-s − 0.571·21-s − 1.28·23-s − 0.988·25-s − 1.05·27-s − 0.959·29-s − 1.49·31-s − 2.15·33-s − 0.0382·35-s − 1.83·37-s + 0.527·39-s + 0.485·41-s + 0.844·43-s − 0.179·45-s + 1.28·47-s − 0.876·49-s − 2.53·51-s + 1.41·53-s − 0.144·55-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 \( 1 + 2.81T + 3T^{2} \)
5 \( 1 + 0.243T + 5T^{2} \)
7 \( 1 - 0.929T + 7T^{2} \)
11 \( 1 - 4.38T + 11T^{2} \)
13 \( 1 + 1.17T + 13T^{2} \)
17 \( 1 - 6.41T + 17T^{2} \)
23 \( 1 + 6.17T + 23T^{2} \)
29 \( 1 + 5.16T + 29T^{2} \)
31 \( 1 + 8.29T + 31T^{2} \)
37 \( 1 + 11.1T + 37T^{2} \)
41 \( 1 - 3.11T + 41T^{2} \)
43 \( 1 - 5.53T + 43T^{2} \)
47 \( 1 - 8.81T + 47T^{2} \)
53 \( 1 - 10.2T + 53T^{2} \)
59 \( 1 + 11.0T + 59T^{2} \)
61 \( 1 - 0.358T + 61T^{2} \)
67 \( 1 - 11.8T + 67T^{2} \)
71 \( 1 - 7.27T + 71T^{2} \)
73 \( 1 + 14.0T + 73T^{2} \)
83 \( 1 - 6.31T + 83T^{2} \)
89 \( 1 - 5.87T + 89T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−7.40753379975398321351853910763, −7.08218425561319810523555752008, −5.98658406271656252853507528014, −5.73919214419486065027945854583, −5.06016527215786288619040639727, −4.03837796606622158170059979457, −3.63563550499614821659707145878, −1.94825563107722273471001142749, −1.17074265416223335376936033655, 0, 1.17074265416223335376936033655, 1.94825563107722273471001142749, 3.63563550499614821659707145878, 4.03837796606622158170059979457, 5.06016527215786288619040639727, 5.73919214419486065027945854583, 5.98658406271656252853507528014, 7.08218425561319810523555752008, 7.40753379975398321351853910763

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