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  − 1.76·2-s + 1.11·4-s + 2.54·5-s − 4.97·7-s + 1.56·8-s − 4.49·10-s − 0.283·11-s − 0.170·13-s + 8.78·14-s − 4.98·16-s − 5.92·17-s + 4.65·19-s + 2.84·20-s + 0.499·22-s − 23-s + 1.49·25-s + 0.301·26-s − 5.54·28-s + 29-s + 8.96·31-s + 5.67·32-s + 10.4·34-s − 12.6·35-s + 0.201·37-s − 8.22·38-s + 3.98·40-s + 6.77·41-s + ⋯
L(s)  = 1  − 1.24·2-s + 0.557·4-s + 1.13·5-s − 1.88·7-s + 0.552·8-s − 1.42·10-s − 0.0853·11-s − 0.0474·13-s + 2.34·14-s − 1.24·16-s − 1.43·17-s + 1.06·19-s + 0.635·20-s + 0.106·22-s − 0.208·23-s + 0.298·25-s + 0.0591·26-s − 1.04·28-s + 0.185·29-s + 1.61·31-s + 1.00·32-s + 1.79·34-s − 2.14·35-s + 0.0331·37-s − 1.33·38-s + 0.629·40-s + 1.05·41-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 \( 1 + 1.76T + 2T^{2} \)
5 \( 1 - 2.54T + 5T^{2} \)
7 \( 1 + 4.97T + 7T^{2} \)
11 \( 1 + 0.283T + 11T^{2} \)
13 \( 1 + 0.170T + 13T^{2} \)
17 \( 1 + 5.92T + 17T^{2} \)
19 \( 1 - 4.65T + 19T^{2} \)
31 \( 1 - 8.96T + 31T^{2} \)
37 \( 1 - 0.201T + 37T^{2} \)
41 \( 1 - 6.77T + 41T^{2} \)
43 \( 1 + 2.97T + 43T^{2} \)
47 \( 1 + 7.64T + 47T^{2} \)
53 \( 1 + 2.22T + 53T^{2} \)
59 \( 1 - 15.2T + 59T^{2} \)
61 \( 1 - 1.08T + 61T^{2} \)
67 \( 1 + 4.78T + 67T^{2} \)
71 \( 1 + 15.8T + 71T^{2} \)
73 \( 1 - 5.09T + 73T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 - 4.95T + 83T^{2} \)
89 \( 1 - 3.87T + 89T^{2} \)
97 \( 1 + 16.0T + 97T^{2} \)
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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

−7.84998339031698467917760717670, −6.90552820707660723225675266055, −6.54602615595669100544365517917, −5.89384027114067860670580667860, −4.93340714671671356968952052999, −3.95320304461489490176058330073, −2.85447807563427750916350283076, −2.24772796489840050780177183489, −1.06848718877325941063721111358, 0, 1.06848718877325941063721111358, 2.24772796489840050780177183489, 2.85447807563427750916350283076, 3.95320304461489490176058330073, 4.93340714671671356968952052999, 5.89384027114067860670580667860, 6.54602615595669100544365517917, 6.90552820707660723225675266055, 7.84998339031698467917760717670

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