Properties

Label 2-503-1.1-c1-0-24
Degree $2$
Conductor $503$
Sign $1$
Analytic cond. $4.01647$
Root an. cond. $2.00411$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3·3-s − 4-s − 2·5-s + 3·6-s + 3·7-s − 3·8-s + 6·9-s − 2·10-s + 3·11-s − 3·12-s + 5·13-s + 3·14-s − 6·15-s − 16-s − 8·17-s + 6·18-s + 4·19-s + 2·20-s + 9·21-s + 3·22-s − 5·23-s − 9·24-s − 25-s + 5·26-s + 9·27-s − 3·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s − 1/2·4-s − 0.894·5-s + 1.22·6-s + 1.13·7-s − 1.06·8-s + 2·9-s − 0.632·10-s + 0.904·11-s − 0.866·12-s + 1.38·13-s + 0.801·14-s − 1.54·15-s − 1/4·16-s − 1.94·17-s + 1.41·18-s + 0.917·19-s + 0.447·20-s + 1.96·21-s + 0.639·22-s − 1.04·23-s − 1.83·24-s − 1/5·25-s + 0.980·26-s + 1.73·27-s − 0.566·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(503\)
Sign: $1$
Analytic conductor: \(4.01647\)
Root analytic conductor: \(2.00411\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 503,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.918896622\)
\(L(\frac12)\) \(\approx\) \(2.918896622\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad503 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 - p T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−11.22003472564326827475162641185, −9.707744155318615349866592838938, −8.743550895982608798765924402380, −8.477653816930119341239566153030, −7.61354678640816137542686480627, −6.34758581113783796961367002258, −4.68232999229458345747448107651, −4.01103324172883747601833367586, −3.36525387639233754741603332272, −1.77760561249637244183542914407, 1.77760561249637244183542914407, 3.36525387639233754741603332272, 4.01103324172883747601833367586, 4.68232999229458345747448107651, 6.34758581113783796961367002258, 7.61354678640816137542686480627, 8.477653816930119341239566153030, 8.743550895982608798765924402380, 9.707744155318615349866592838938, 11.22003472564326827475162641185

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