Properties

Label 2-1003-1.1-c1-0-19
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.79·2-s − 2.09·3-s + 1.23·4-s − 2.26·5-s + 3.76·6-s − 0.408·7-s + 1.38·8-s + 1.37·9-s + 4.07·10-s − 5.30·11-s − 2.57·12-s + 4.40·13-s + 0.733·14-s + 4.74·15-s − 4.94·16-s + 17-s − 2.48·18-s + 6.57·19-s − 2.79·20-s + 0.854·21-s + 9.54·22-s − 3.22·23-s − 2.89·24-s + 0.149·25-s − 7.91·26-s + 3.39·27-s − 0.502·28-s + ⋯
L(s)  = 1  − 1.27·2-s − 1.20·3-s + 0.615·4-s − 1.01·5-s + 1.53·6-s − 0.154·7-s + 0.489·8-s + 0.459·9-s + 1.28·10-s − 1.60·11-s − 0.743·12-s + 1.22·13-s + 0.196·14-s + 1.22·15-s − 1.23·16-s + 0.242·17-s − 0.584·18-s + 1.50·19-s − 0.624·20-s + 0.186·21-s + 2.03·22-s − 0.672·23-s − 0.591·24-s + 0.0299·25-s − 1.55·26-s + 0.652·27-s − 0.0948·28-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: \(1\)
Selberg data: \((2,\ 1003,\ (\ :1/2),\ -1)\)

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)$
bad17 \( 1 - T \)
59 \( 1 - T \)
good2 \( 1 + 1.79T + 2T^{2} \)
3 \( 1 + 2.09T + 3T^{2} \)
5 \( 1 + 2.26T + 5T^{2} \)
7 \( 1 + 0.408T + 7T^{2} \)
11 \( 1 + 5.30T + 11T^{2} \)
13 \( 1 - 4.40T + 13T^{2} \)
19 \( 1 - 6.57T + 19T^{2} \)
23 \( 1 + 3.22T + 23T^{2} \)
29 \( 1 - 5.31T + 29T^{2} \)
31 \( 1 - 8.06T + 31T^{2} \)
37 \( 1 + 7.46T + 37T^{2} \)
41 \( 1 - 0.557T + 41T^{2} \)
43 \( 1 - 7.66T + 43T^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 + 13.4T + 53T^{2} \)
61 \( 1 - 2.66T + 61T^{2} \)
67 \( 1 + 7.99T + 67T^{2} \)
71 \( 1 + 6.23T + 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 - 10.6T + 79T^{2} \)
83 \( 1 + 7.40T + 83T^{2} \)
89 \( 1 - 7.94T + 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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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

−9.715728359767347441782445969410, −8.548216921106851000617913296877, −7.969987027933019075574558383904, −7.33468104826672007804363005808, −6.23264083266056144157513498667, −5.31464171667771381221124108474, −4.38107156479392867426412648949, −3.00091463121080336648513515386, −1.06951894810323582306156696495, 0, 1.06951894810323582306156696495, 3.00091463121080336648513515386, 4.38107156479392867426412648949, 5.31464171667771381221124108474, 6.23264083266056144157513498667, 7.33468104826672007804363005808, 7.969987027933019075574558383904, 8.548216921106851000617913296877, 9.715728359767347441782445969410

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