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  + 0.467·2-s − 1.78·4-s + 0.0105·5-s − 1.85·7-s − 1.76·8-s + 0.00492·10-s + 1.01·11-s + 2.69·13-s − 0.868·14-s + 2.73·16-s + 3.50·17-s − 7.48·19-s − 0.0187·20-s + 0.472·22-s − 23-s − 4.99·25-s + 1.26·26-s + 3.31·28-s + 29-s + 9.06·31-s + 4.81·32-s + 1.63·34-s − 0.0195·35-s − 3.80·37-s − 3.49·38-s − 0.0186·40-s − 0.921·41-s + ⋯
L(s)  = 1  + 0.330·2-s − 0.890·4-s + 0.00471·5-s − 0.702·7-s − 0.624·8-s + 0.00155·10-s + 0.305·11-s + 0.748·13-s − 0.232·14-s + 0.684·16-s + 0.851·17-s − 1.71·19-s − 0.00420·20-s + 0.100·22-s − 0.208·23-s − 0.999·25-s + 0.247·26-s + 0.626·28-s + 0.185·29-s + 1.62·31-s + 0.850·32-s + 0.281·34-s − 0.00331·35-s − 0.625·37-s − 0.566·38-s − 0.00294·40-s − 0.143·41-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 \( 1 - 0.467T + 2T^{2} \)
5 \( 1 - 0.0105T + 5T^{2} \)
7 \( 1 + 1.85T + 7T^{2} \)
11 \( 1 - 1.01T + 11T^{2} \)
13 \( 1 - 2.69T + 13T^{2} \)
17 \( 1 - 3.50T + 17T^{2} \)
19 \( 1 + 7.48T + 19T^{2} \)
31 \( 1 - 9.06T + 31T^{2} \)
37 \( 1 + 3.80T + 37T^{2} \)
41 \( 1 + 0.921T + 41T^{2} \)
43 \( 1 - 5.98T + 43T^{2} \)
47 \( 1 - 8.75T + 47T^{2} \)
53 \( 1 - 9.78T + 53T^{2} \)
59 \( 1 + 13.5T + 59T^{2} \)
61 \( 1 + 1.50T + 61T^{2} \)
67 \( 1 + 7.98T + 67T^{2} \)
71 \( 1 + 3.19T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 - 9.57T + 79T^{2} \)
83 \( 1 + 15.0T + 83T^{2} \)
89 \( 1 + 9.07T + 89T^{2} \)
97 \( 1 - 1.28T + 97T^{2} \)
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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

−7.916364920955223046725467937541, −6.85648915517481861237893870138, −6.06495889541878464925401047492, −5.78600292863150342650482017475, −4.64611689002030465574977635936, −4.06070050795120900391071907274, −3.44726705868363189815488962538, −2.50488563267297068479498272611, −1.18090658188474038834695081915, 0, 1.18090658188474038834695081915, 2.50488563267297068479498272611, 3.44726705868363189815488962538, 4.06070050795120900391071907274, 4.64611689002030465574977635936, 5.78600292863150342650482017475, 6.06495889541878464925401047492, 6.85648915517481861237893870138, 7.916364920955223046725467937541

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