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  + 2.82·5-s + 4.85·7-s − 3·9-s + 1.69·11-s + 3.33·13-s − 2.21·17-s + 19-s + 5.35·23-s + 2.96·25-s − 6.61·29-s + 1.16·31-s + 13.7·35-s + 4.72·37-s − 3.45·41-s − 3.96·43-s − 8.46·45-s + 0.312·47-s + 16.6·49-s + 5.80·53-s + 4.77·55-s + 6.38·59-s + 3.99·61-s − 14.5·63-s + 9.42·65-s − 1.03·67-s + 15.9·71-s − 16.0·73-s + ⋯
L(s)  = 1  + 1.26·5-s + 1.83·7-s − 9-s + 0.510·11-s + 0.926·13-s − 0.536·17-s + 0.229·19-s + 1.11·23-s + 0.592·25-s − 1.22·29-s + 0.208·31-s + 2.31·35-s + 0.777·37-s − 0.540·41-s − 0.605·43-s − 1.26·45-s + 0.0456·47-s + 2.37·49-s + 0.797·53-s + 0.644·55-s + 0.831·59-s + 0.511·61-s − 1.83·63-s + 1.16·65-s − 0.126·67-s + 1.89·71-s − 1.87·73-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  =  $(2,\ 6004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.447752869$
$L(\frac12)$  $\approx$  $3.447752869$
$L(\frac{3}{2})$   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 \( 1 - 2.82T + 5T^{2} \)
7 \( 1 - 4.85T + 7T^{2} \)
11 \( 1 - 1.69T + 11T^{2} \)
13 \( 1 - 3.33T + 13T^{2} \)
17 \( 1 + 2.21T + 17T^{2} \)
23 \( 1 - 5.35T + 23T^{2} \)
29 \( 1 + 6.61T + 29T^{2} \)
31 \( 1 - 1.16T + 31T^{2} \)
37 \( 1 - 4.72T + 37T^{2} \)
41 \( 1 + 3.45T + 41T^{2} \)
43 \( 1 + 3.96T + 43T^{2} \)
47 \( 1 - 0.312T + 47T^{2} \)
53 \( 1 - 5.80T + 53T^{2} \)
59 \( 1 - 6.38T + 59T^{2} \)
61 \( 1 - 3.99T + 61T^{2} \)
67 \( 1 + 1.03T + 67T^{2} \)
71 \( 1 - 15.9T + 71T^{2} \)
73 \( 1 + 16.0T + 73T^{2} \)
83 \( 1 + 1.04T + 83T^{2} \)
89 \( 1 + 16.2T + 89T^{2} \)
97 \( 1 + 6.30T + 97T^{2} \)
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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

−8.303274379251654943750285660033, −7.39142540645293229808107911355, −6.56958969100160247023131840551, −5.74565643251285217405851440482, −5.37721340914705632393773713355, −4.61772912687028466078159975575, −3.66475173694483580133822161583, −2.53632258382607028270638653778, −1.82056650467146323526691565254, −1.07198056032298388612842116247, 1.07198056032298388612842116247, 1.82056650467146323526691565254, 2.53632258382607028270638653778, 3.66475173694483580133822161583, 4.61772912687028466078159975575, 5.37721340914705632393773713355, 5.74565643251285217405851440482, 6.56958969100160247023131840551, 7.39142540645293229808107911355, 8.303274379251654943750285660033

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