Properties

Label 2-1003-1.1-c1-0-11
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.25·2-s − 2.18·3-s − 0.419·4-s + 4.45·5-s + 2.74·6-s − 4.53·7-s + 3.04·8-s + 1.77·9-s − 5.60·10-s − 0.915·11-s + 0.917·12-s + 3.82·13-s + 5.69·14-s − 9.73·15-s − 2.98·16-s − 17-s − 2.23·18-s + 1.42·19-s − 1.87·20-s + 9.90·21-s + 1.15·22-s − 0.780·23-s − 6.64·24-s + 14.8·25-s − 4.80·26-s + 2.67·27-s + 1.90·28-s + ⋯
L(s)  = 1  − 0.888·2-s − 1.26·3-s − 0.209·4-s + 1.99·5-s + 1.12·6-s − 1.71·7-s + 1.07·8-s + 0.591·9-s − 1.77·10-s − 0.276·11-s + 0.264·12-s + 1.06·13-s + 1.52·14-s − 2.51·15-s − 0.746·16-s − 0.242·17-s − 0.525·18-s + 0.327·19-s − 0.418·20-s + 2.16·21-s + 0.245·22-s − 0.162·23-s − 1.35·24-s + 2.97·25-s − 0.942·26-s + 0.515·27-s + 0.359·28-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

\(L(1)\) \(\approx\) \(0.5946641401\)
\(L(\frac12)\) \(\approx\) \(0.5946641401\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.25T + 2T^{2} \)
3 \( 1 + 2.18T + 3T^{2} \)
5 \( 1 - 4.45T + 5T^{2} \)
7 \( 1 + 4.53T + 7T^{2} \)
11 \( 1 + 0.915T + 11T^{2} \)
13 \( 1 - 3.82T + 13T^{2} \)
19 \( 1 - 1.42T + 19T^{2} \)
23 \( 1 + 0.780T + 23T^{2} \)
29 \( 1 + 6.11T + 29T^{2} \)
31 \( 1 + 9.72T + 31T^{2} \)
37 \( 1 + 7.06T + 37T^{2} \)
41 \( 1 + 2.81T + 41T^{2} \)
43 \( 1 - 9.73T + 43T^{2} \)
47 \( 1 - 7.06T + 47T^{2} \)
53 \( 1 - 1.70T + 53T^{2} \)
61 \( 1 - 14.8T + 61T^{2} \)
67 \( 1 - 2.99T + 67T^{2} \)
71 \( 1 - 7.46T + 71T^{2} \)
73 \( 1 + 1.83T + 73T^{2} \)
79 \( 1 - 7.75T + 79T^{2} \)
83 \( 1 - 9.33T + 83T^{2} \)
89 \( 1 - 3.95T + 89T^{2} \)
97 \( 1 - 1.94T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.998573986212528646112670703884, −9.214769863436551967753815177838, −8.880940969579053544300017897944, −7.17363853146481787768687165640, −6.46317496411785414008912838623, −5.72463823865093981038831541056, −5.28224623131527969756597005193, −3.64094641572084543362986909079, −2.08427766360933779650479689036, −0.72950024748434165125500007565, 0.72950024748434165125500007565, 2.08427766360933779650479689036, 3.64094641572084543362986909079, 5.28224623131527969756597005193, 5.72463823865093981038831541056, 6.46317496411785414008912838623, 7.17363853146481787768687165640, 8.880940969579053544300017897944, 9.214769863436551967753815177838, 9.998573986212528646112670703884

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