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  − 0.654·5-s − 2.24·7-s − 3·9-s + 4.24·11-s − 4.30·13-s − 4.04·17-s + 19-s − 2.93·23-s − 4.57·25-s + 1.01·29-s + 8.52·31-s + 1.47·35-s − 8.68·37-s − 9.95·41-s + 4.61·43-s + 1.96·45-s + 9.53·47-s − 1.95·49-s + 3.71·53-s − 2.77·55-s + 0.335·59-s + 7.24·61-s + 6.73·63-s + 2.82·65-s + 0.613·67-s + 7.70·71-s + 1.41·73-s + ⋯
L(s)  = 1  − 0.292·5-s − 0.849·7-s − 9-s + 1.27·11-s − 1.19·13-s − 0.980·17-s + 0.229·19-s − 0.612·23-s − 0.914·25-s + 0.189·29-s + 1.53·31-s + 0.248·35-s − 1.42·37-s − 1.55·41-s + 0.704·43-s + 0.292·45-s + 1.39·47-s − 0.279·49-s + 0.510·53-s − 0.374·55-s + 0.0436·59-s + 0.927·61-s + 0.849·63-s + 0.349·65-s + 0.0749·67-s + 0.914·71-s + 0.165·73-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  =  $(2,\ 6004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9209720648$
$L(\frac12)$  $\approx$  $0.9209720648$
$L(\frac{3}{2})$   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 \( 1 + 0.654T + 5T^{2} \)
7 \( 1 + 2.24T + 7T^{2} \)
11 \( 1 - 4.24T + 11T^{2} \)
13 \( 1 + 4.30T + 13T^{2} \)
17 \( 1 + 4.04T + 17T^{2} \)
23 \( 1 + 2.93T + 23T^{2} \)
29 \( 1 - 1.01T + 29T^{2} \)
31 \( 1 - 8.52T + 31T^{2} \)
37 \( 1 + 8.68T + 37T^{2} \)
41 \( 1 + 9.95T + 41T^{2} \)
43 \( 1 - 4.61T + 43T^{2} \)
47 \( 1 - 9.53T + 47T^{2} \)
53 \( 1 - 3.71T + 53T^{2} \)
59 \( 1 - 0.335T + 59T^{2} \)
61 \( 1 - 7.24T + 61T^{2} \)
67 \( 1 - 0.613T + 67T^{2} \)
71 \( 1 - 7.70T + 71T^{2} \)
73 \( 1 - 1.41T + 73T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 + 2.60T + 89T^{2} \)
97 \( 1 - 12.5T + 97T^{2} \)
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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

−8.237176387069559077007572999350, −7.19549588234557851364928849075, −6.69794966312522662210442286512, −6.05958903882889024365558942152, −5.24872415462556443298037733023, −4.33504871954382116236423492124, −3.65264613439003895154927687318, −2.81361008596563582629856639807, −1.98813996298801293998505549488, −0.47813055023363177415163328192, 0.47813055023363177415163328192, 1.98813996298801293998505549488, 2.81361008596563582629856639807, 3.65264613439003895154927687318, 4.33504871954382116236423492124, 5.24872415462556443298037733023, 6.05958903882889024365558942152, 6.69794966312522662210442286512, 7.19549588234557851364928849075, 8.237176387069559077007572999350

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