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  − 2.72·2-s + 5.42·4-s + 1.02·5-s + 4.32·7-s − 9.33·8-s − 2.79·10-s − 2.38·11-s + 0.754·13-s − 11.7·14-s + 14.5·16-s + 1.03·17-s − 8.17·19-s + 5.55·20-s + 6.49·22-s − 23-s − 3.94·25-s − 2.05·26-s + 23.4·28-s + 29-s − 5.92·31-s − 21.0·32-s − 2.83·34-s + 4.43·35-s + 3.05·37-s + 22.2·38-s − 9.56·40-s − 1.01·41-s + ⋯
L(s)  = 1  − 1.92·2-s + 2.71·4-s + 0.458·5-s + 1.63·7-s − 3.29·8-s − 0.883·10-s − 0.718·11-s + 0.209·13-s − 3.15·14-s + 3.64·16-s + 0.252·17-s − 1.87·19-s + 1.24·20-s + 1.38·22-s − 0.208·23-s − 0.789·25-s − 0.403·26-s + 4.43·28-s + 0.185·29-s − 1.06·31-s − 3.72·32-s − 0.485·34-s + 0.749·35-s + 0.502·37-s + 3.61·38-s − 1.51·40-s − 0.157·41-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 \( 1 + 2.72T + 2T^{2} \)
5 \( 1 - 1.02T + 5T^{2} \)
7 \( 1 - 4.32T + 7T^{2} \)
11 \( 1 + 2.38T + 11T^{2} \)
13 \( 1 - 0.754T + 13T^{2} \)
17 \( 1 - 1.03T + 17T^{2} \)
19 \( 1 + 8.17T + 19T^{2} \)
31 \( 1 + 5.92T + 31T^{2} \)
37 \( 1 - 3.05T + 37T^{2} \)
41 \( 1 + 1.01T + 41T^{2} \)
43 \( 1 - 8.18T + 43T^{2} \)
47 \( 1 + 8.08T + 47T^{2} \)
53 \( 1 - 0.478T + 53T^{2} \)
59 \( 1 - 8.09T + 59T^{2} \)
61 \( 1 + 14.8T + 61T^{2} \)
67 \( 1 + 0.830T + 67T^{2} \)
71 \( 1 - 1.79T + 71T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 - 2.46T + 79T^{2} \)
83 \( 1 + 4.46T + 83T^{2} \)
89 \( 1 - 5.57T + 89T^{2} \)
97 \( 1 + 0.684T + 97T^{2} \)
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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

−7.940604323566510964100723481576, −7.45076175010002191995208592825, −6.50340252995891140331434842657, −5.87142279557795411843466875034, −5.05839491533456497457149951566, −3.94601811844858903433690594714, −2.53545760016929561265167301832, −2.01234635076835686333223910967, −1.31300442791953564075274872658, 0, 1.31300442791953564075274872658, 2.01234635076835686333223910967, 2.53545760016929561265167301832, 3.94601811844858903433690594714, 5.05839491533456497457149951566, 5.87142279557795411843466875034, 6.50340252995891140331434842657, 7.45076175010002191995208592825, 7.940604323566510964100723481576

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