Properties

Degree $2$
Conductor $6003$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.96·2-s + 1.87·4-s + 3.90·5-s − 0.839·7-s − 0.255·8-s + 7.67·10-s − 4.08·11-s − 6.57·13-s − 1.65·14-s − 4.24·16-s − 0.379·17-s − 6.25·19-s + 7.29·20-s − 8.03·22-s − 23-s + 10.2·25-s − 12.9·26-s − 1.57·28-s + 29-s + 3.23·31-s − 7.83·32-s − 0.745·34-s − 3.27·35-s − 7.47·37-s − 12.3·38-s − 0.996·40-s + 9.84·41-s + ⋯
L(s)  = 1  + 1.39·2-s + 0.935·4-s + 1.74·5-s − 0.317·7-s − 0.0902·8-s + 2.42·10-s − 1.23·11-s − 1.82·13-s − 0.441·14-s − 1.06·16-s − 0.0919·17-s − 1.43·19-s + 1.63·20-s − 1.71·22-s − 0.208·23-s + 2.04·25-s − 2.53·26-s − 0.296·28-s + 0.185·29-s + 0.580·31-s − 1.38·32-s − 0.127·34-s − 0.553·35-s − 1.22·37-s − 1.99·38-s − 0.157·40-s + 1.53·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{6003} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6003,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 + T \)
29 \( 1 - T \)
good2 \( 1 - 1.96T + 2T^{2} \)
5 \( 1 - 3.90T + 5T^{2} \)
7 \( 1 + 0.839T + 7T^{2} \)
11 \( 1 + 4.08T + 11T^{2} \)
13 \( 1 + 6.57T + 13T^{2} \)
17 \( 1 + 0.379T + 17T^{2} \)
19 \( 1 + 6.25T + 19T^{2} \)
31 \( 1 - 3.23T + 31T^{2} \)
37 \( 1 + 7.47T + 37T^{2} \)
41 \( 1 - 9.84T + 41T^{2} \)
43 \( 1 + 6.99T + 43T^{2} \)
47 \( 1 - 9.65T + 47T^{2} \)
53 \( 1 + 10.9T + 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 - 2.37T + 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 + 4.24T + 71T^{2} \)
73 \( 1 + 8.99T + 73T^{2} \)
79 \( 1 - 1.79T + 79T^{2} \)
83 \( 1 - 6.12T + 83T^{2} \)
89 \( 1 + 4.65T + 89T^{2} \)
97 \( 1 + 2.57T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.40595467774190672930620796411, −6.68113714593534994288166773408, −6.08628015563640792397685435191, −5.46683879282598168141199388264, −4.92522535489232868963233113692, −4.35883554820277107718365662273, −3.02919941313424737233698953556, −2.49351217249105565231915563362, −1.97584307673730374560100598883, 0, 1.97584307673730374560100598883, 2.49351217249105565231915563362, 3.02919941313424737233698953556, 4.35883554820277107718365662273, 4.92522535489232868963233113692, 5.46683879282598168141199388264, 6.08628015563640792397685435191, 6.68113714593534994288166773408, 7.40595467774190672930620796411

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