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  − 2.49·5-s + 4.08·7-s − 3·9-s − 6.35·11-s − 3.31·13-s − 0.507·17-s + 19-s + 3.01·23-s + 1.23·25-s + 4.18·29-s − 9.86·31-s − 10.1·35-s − 9.92·37-s + 0.808·41-s − 5.39·43-s + 7.49·45-s + 10.4·47-s + 9.68·49-s + 10.9·53-s + 15.8·55-s + 3.91·59-s − 8.02·61-s − 12.2·63-s + 8.27·65-s + 5.98·67-s + 10.8·71-s + 0.522·73-s + ⋯
L(s)  = 1  − 1.11·5-s + 1.54·7-s − 9-s − 1.91·11-s − 0.919·13-s − 0.123·17-s + 0.229·19-s + 0.628·23-s + 0.246·25-s + 0.776·29-s − 1.77·31-s − 1.72·35-s − 1.63·37-s + 0.126·41-s − 0.823·43-s + 1.11·45-s + 1.53·47-s + 1.38·49-s + 1.50·53-s + 2.13·55-s + 0.509·59-s − 1.02·61-s − 1.54·63-s + 1.02·65-s + 0.731·67-s + 1.29·71-s + 0.0612·73-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  =  $(2,\ 6004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8749237154$
$L(\frac12)$  $\approx$  $0.8749237154$
$L(\frac{3}{2})$   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 \( 1 + 2.49T + 5T^{2} \)
7 \( 1 - 4.08T + 7T^{2} \)
11 \( 1 + 6.35T + 11T^{2} \)
13 \( 1 + 3.31T + 13T^{2} \)
17 \( 1 + 0.507T + 17T^{2} \)
23 \( 1 - 3.01T + 23T^{2} \)
29 \( 1 - 4.18T + 29T^{2} \)
31 \( 1 + 9.86T + 31T^{2} \)
37 \( 1 + 9.92T + 37T^{2} \)
41 \( 1 - 0.808T + 41T^{2} \)
43 \( 1 + 5.39T + 43T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 - 10.9T + 53T^{2} \)
59 \( 1 - 3.91T + 59T^{2} \)
61 \( 1 + 8.02T + 61T^{2} \)
67 \( 1 - 5.98T + 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 - 0.522T + 73T^{2} \)
83 \( 1 - 2.67T + 83T^{2} \)
89 \( 1 - 1.13T + 89T^{2} \)
97 \( 1 + 7.68T + 97T^{2} \)
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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

−8.161208617629890003905586506524, −7.46402643251315561912691883138, −7.05669079779273034999048185253, −5.43704184899738425912405816356, −5.32857557375624511698157757428, −4.61231508184201739135272509323, −3.63722530991498157911466387694, −2.72522191933860605380199858900, −2.01934882041315541065709682890, −0.46562500952041516561778198642, 0.46562500952041516561778198642, 2.01934882041315541065709682890, 2.72522191933860605380199858900, 3.63722530991498157911466387694, 4.61231508184201739135272509323, 5.32857557375624511698157757428, 5.43704184899738425912405816356, 7.05669079779273034999048185253, 7.46402643251315561912691883138, 8.161208617629890003905586506524

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