Properties

Degree $2$
Conductor $8003$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.40·2-s + 2.42·3-s + 3.77·4-s + 1.57·5-s − 5.82·6-s − 3.75·7-s − 4.27·8-s + 2.86·9-s − 3.79·10-s + 2.83·11-s + 9.15·12-s + 2.38·13-s + 9.01·14-s + 3.82·15-s + 2.71·16-s − 1.20·17-s − 6.89·18-s − 2.50·19-s + 5.96·20-s − 9.08·21-s − 6.82·22-s − 6.40·23-s − 10.3·24-s − 2.50·25-s − 5.74·26-s − 0.322·27-s − 14.1·28-s + ⋯
L(s)  = 1  − 1.69·2-s + 1.39·3-s + 1.88·4-s + 0.705·5-s − 2.37·6-s − 1.41·7-s − 1.51·8-s + 0.955·9-s − 1.19·10-s + 0.856·11-s + 2.64·12-s + 0.662·13-s + 2.40·14-s + 0.987·15-s + 0.678·16-s − 0.292·17-s − 1.62·18-s − 0.574·19-s + 1.33·20-s − 1.98·21-s − 1.45·22-s − 1.33·23-s − 2.11·24-s − 0.501·25-s − 1.12·26-s − 0.0621·27-s − 2.67·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8003\)    =    \(53 \cdot 151\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{8003} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8003,\ (\ :1/2),\ -1)\)

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)$
bad53 \( 1 + T \)
151 \( 1 + T \)
good2 \( 1 + 2.40T + 2T^{2} \)
3 \( 1 - 2.42T + 3T^{2} \)
5 \( 1 - 1.57T + 5T^{2} \)
7 \( 1 + 3.75T + 7T^{2} \)
11 \( 1 - 2.83T + 11T^{2} \)
13 \( 1 - 2.38T + 13T^{2} \)
17 \( 1 + 1.20T + 17T^{2} \)
19 \( 1 + 2.50T + 19T^{2} \)
23 \( 1 + 6.40T + 23T^{2} \)
29 \( 1 - 5.21T + 29T^{2} \)
31 \( 1 - 6.95T + 31T^{2} \)
37 \( 1 + 2.27T + 37T^{2} \)
41 \( 1 + 3.13T + 41T^{2} \)
43 \( 1 + 1.85T + 43T^{2} \)
47 \( 1 + 5.28T + 47T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 + 4.26T + 61T^{2} \)
67 \( 1 - 7.32T + 67T^{2} \)
71 \( 1 - 2.39T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 + 6.98T + 83T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 + 8.99T + 97T^{2} \)
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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

−7.84903677856839200049355631429, −6.89339666085820440958663757674, −6.42018701267152378672105440847, −5.98936098258996787945422400500, −4.34281319912290541500896835949, −3.51248519358082656901147243467, −2.80157629881630287636951250161, −2.07445967560423732265752026654, −1.35512972405529072124820713518, 0, 1.35512972405529072124820713518, 2.07445967560423732265752026654, 2.80157629881630287636951250161, 3.51248519358082656901147243467, 4.34281319912290541500896835949, 5.98936098258996787945422400500, 6.42018701267152378672105440847, 6.89339666085820440958663757674, 7.84903677856839200049355631429

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