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  + 2.51·2-s + 4.30·4-s − 2.07·5-s − 0.329·7-s + 5.79·8-s − 5.20·10-s − 4.89·11-s + 2.21·13-s − 0.827·14-s + 5.93·16-s − 4.14·17-s + 0.434·19-s − 8.93·20-s − 12.2·22-s − 23-s − 0.696·25-s + 5.56·26-s − 1.41·28-s + 29-s + 11.0·31-s + 3.31·32-s − 10.4·34-s + 0.683·35-s + 0.655·37-s + 1.09·38-s − 12.0·40-s − 7.37·41-s + ⋯
L(s)  = 1  + 1.77·2-s + 2.15·4-s − 0.927·5-s − 0.124·7-s + 2.04·8-s − 1.64·10-s − 1.47·11-s + 0.614·13-s − 0.221·14-s + 1.48·16-s − 1.00·17-s + 0.0997·19-s − 1.99·20-s − 2.62·22-s − 0.208·23-s − 0.139·25-s + 1.09·26-s − 0.268·28-s + 0.185·29-s + 1.97·31-s + 0.585·32-s − 1.78·34-s + 0.115·35-s + 0.107·37-s + 0.177·38-s − 1.89·40-s − 1.15·41-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 \( 1 - 2.51T + 2T^{2} \)
5 \( 1 + 2.07T + 5T^{2} \)
7 \( 1 + 0.329T + 7T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 - 2.21T + 13T^{2} \)
17 \( 1 + 4.14T + 17T^{2} \)
19 \( 1 - 0.434T + 19T^{2} \)
31 \( 1 - 11.0T + 31T^{2} \)
37 \( 1 - 0.655T + 37T^{2} \)
41 \( 1 + 7.37T + 41T^{2} \)
43 \( 1 - 0.463T + 43T^{2} \)
47 \( 1 + 4.14T + 47T^{2} \)
53 \( 1 + 2.30T + 53T^{2} \)
59 \( 1 + 9.22T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 + 14.6T + 67T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 - 7.93T + 73T^{2} \)
79 \( 1 + 6.32T + 79T^{2} \)
83 \( 1 - 16.4T + 83T^{2} \)
89 \( 1 + 7.88T + 89T^{2} \)
97 \( 1 + 15.8T + 97T^{2} \)
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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

−7.70499628160858140432467463499, −6.68311059335130795657009976203, −6.24396311752168171425515571184, −5.40500971740901990841100348351, −4.58893665506334697279614727993, −4.32316130589020553861534162848, −3.20054694225260652422093602557, −2.88663363061383700615911942346, −1.76668711289421790111450920079, 0, 1.76668711289421790111450920079, 2.88663363061383700615911942346, 3.20054694225260652422093602557, 4.32316130589020553861534162848, 4.58893665506334697279614727993, 5.40500971740901990841100348351, 6.24396311752168171425515571184, 6.68311059335130795657009976203, 7.70499628160858140432467463499

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