Properties

Label 2-1003-1.1-c1-0-44
Degree $2$
Conductor $1003$
Sign $1$
Analytic cond. $8.00899$
Root an. cond. $2.83001$
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  + 2.59·2-s − 2.74·3-s + 4.71·4-s + 3.63·5-s − 7.11·6-s + 3.18·7-s + 7.04·8-s + 4.54·9-s + 9.40·10-s − 2.94·11-s − 12.9·12-s − 1.90·13-s + 8.25·14-s − 9.96·15-s + 8.82·16-s − 17-s + 11.7·18-s + 3.08·19-s + 17.1·20-s − 8.74·21-s − 7.63·22-s − 7.87·23-s − 19.3·24-s + 8.17·25-s − 4.94·26-s − 4.23·27-s + 15.0·28-s + ⋯
L(s)  = 1  + 1.83·2-s − 1.58·3-s + 2.35·4-s + 1.62·5-s − 2.90·6-s + 1.20·7-s + 2.49·8-s + 1.51·9-s + 2.97·10-s − 0.888·11-s − 3.74·12-s − 0.528·13-s + 2.20·14-s − 2.57·15-s + 2.20·16-s − 0.242·17-s + 2.77·18-s + 0.707·19-s + 3.82·20-s − 1.90·21-s − 1.62·22-s − 1.64·23-s − 3.94·24-s + 1.63·25-s − 0.968·26-s − 0.814·27-s + 2.83·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1003\)    =    \(17 \cdot 59\)
Sign: $1$
Analytic conductor: \(8.00899\)
Root analytic conductor: \(2.83001\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1003,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.075109465\)
\(L(\frac12)\) \(\approx\) \(4.075109465\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + T \)
59 \( 1 - T \)
good2 \( 1 - 2.59T + 2T^{2} \)
3 \( 1 + 2.74T + 3T^{2} \)
5 \( 1 - 3.63T + 5T^{2} \)
7 \( 1 - 3.18T + 7T^{2} \)
11 \( 1 + 2.94T + 11T^{2} \)
13 \( 1 + 1.90T + 13T^{2} \)
19 \( 1 - 3.08T + 19T^{2} \)
23 \( 1 + 7.87T + 23T^{2} \)
29 \( 1 + 0.623T + 29T^{2} \)
31 \( 1 + 5.23T + 31T^{2} \)
37 \( 1 - 1.13T + 37T^{2} \)
41 \( 1 - 5.79T + 41T^{2} \)
43 \( 1 + 4.51T + 43T^{2} \)
47 \( 1 - 5.03T + 47T^{2} \)
53 \( 1 - 11.8T + 53T^{2} \)
61 \( 1 + 8.78T + 61T^{2} \)
67 \( 1 + 5.43T + 67T^{2} \)
71 \( 1 + 15.0T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 - 14.4T + 83T^{2} \)
89 \( 1 + 7.13T + 89T^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
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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

−10.66013962210804094883114459904, −9.575908839916431661898584919227, −7.81118126584702312872089845043, −6.89676202856989811593882659526, −5.97717279617701437061383151758, −5.49941665401812683891895726733, −5.10079431431854520484441431035, −4.24329450510981642616963628101, −2.48998298991182567311074057867, −1.64806429396503185880595473737, 1.64806429396503185880595473737, 2.48998298991182567311074057867, 4.24329450510981642616963628101, 5.10079431431854520484441431035, 5.49941665401812683891895726733, 5.97717279617701437061383151758, 6.89676202856989811593882659526, 7.81118126584702312872089845043, 9.575908839916431661898584919227, 10.66013962210804094883114459904

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