Properties

Label 2-8003-1.1-c1-0-56
Degree $2$
Conductor $8003$
Sign $1$
Analytic cond. $63.9042$
Root an. cond. $7.99401$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.65·2-s − 0.851·3-s + 5.02·4-s − 0.493·5-s + 2.25·6-s + 4.68·7-s − 8.01·8-s − 2.27·9-s + 1.30·10-s − 6.10·11-s − 4.27·12-s − 1.88·13-s − 12.4·14-s + 0.420·15-s + 11.1·16-s − 0.365·17-s + 6.02·18-s − 1.92·19-s − 2.48·20-s − 3.99·21-s + 16.1·22-s − 5.59·23-s + 6.82·24-s − 4.75·25-s + 5.00·26-s + 4.49·27-s + 23.5·28-s + ⋯
L(s)  = 1  − 1.87·2-s − 0.491·3-s + 2.51·4-s − 0.220·5-s + 0.921·6-s + 1.77·7-s − 2.83·8-s − 0.758·9-s + 0.413·10-s − 1.84·11-s − 1.23·12-s − 0.523·13-s − 3.32·14-s + 0.108·15-s + 2.79·16-s − 0.0886·17-s + 1.42·18-s − 0.441·19-s − 0.554·20-s − 0.871·21-s + 3.44·22-s − 1.16·23-s + 1.39·24-s − 0.951·25-s + 0.981·26-s + 0.864·27-s + 4.45·28-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: $1$
Analytic conductor: \(63.9042\)
Root analytic conductor: \(7.99401\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8003,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1984088198\)
\(L(\frac12)\) \(\approx\) \(0.1984088198\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.65T + 2T^{2} \)
3 \( 1 + 0.851T + 3T^{2} \)
5 \( 1 + 0.493T + 5T^{2} \)
7 \( 1 - 4.68T + 7T^{2} \)
11 \( 1 + 6.10T + 11T^{2} \)
13 \( 1 + 1.88T + 13T^{2} \)
17 \( 1 + 0.365T + 17T^{2} \)
19 \( 1 + 1.92T + 19T^{2} \)
23 \( 1 + 5.59T + 23T^{2} \)
29 \( 1 - 4.25T + 29T^{2} \)
31 \( 1 + 0.539T + 31T^{2} \)
37 \( 1 + 7.84T + 37T^{2} \)
41 \( 1 + 2.57T + 41T^{2} \)
43 \( 1 + 1.65T + 43T^{2} \)
47 \( 1 + 4.53T + 47T^{2} \)
59 \( 1 + 7.19T + 59T^{2} \)
61 \( 1 + 1.86T + 61T^{2} \)
67 \( 1 + 3.62T + 67T^{2} \)
71 \( 1 - 0.316T + 71T^{2} \)
73 \( 1 - 3.71T + 73T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 + 2.32T + 83T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 - 6.51T + 97T^{2} \)
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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

−7.905197423454725987596049847511, −7.64748216990319307385053791649, −6.74537008109600705041422981611, −5.81238166959536932308508241155, −5.28045841120976025511515146427, −4.50680119269449256786516418985, −3.01124275301478453918029458613, −2.22489682119929573282999885556, −1.64228618896093727866011838400, −0.29040438994785939187093594776, 0.29040438994785939187093594776, 1.64228618896093727866011838400, 2.22489682119929573282999885556, 3.01124275301478453918029458613, 4.50680119269449256786516418985, 5.28045841120976025511515146427, 5.81238166959536932308508241155, 6.74537008109600705041422981611, 7.64748216990319307385053791649, 7.905197423454725987596049847511

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