Properties

Degree $2$
Conductor $503$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.0830·2-s + 0.315·3-s − 1.99·4-s + 2.25·5-s + 0.0262·6-s − 3.20·7-s − 0.331·8-s − 2.90·9-s + 0.186·10-s − 0.218·11-s − 0.629·12-s − 4.17·13-s − 0.266·14-s + 0.710·15-s + 3.95·16-s − 4.68·17-s − 0.240·18-s + 3.43·19-s − 4.48·20-s − 1.01·21-s − 0.0181·22-s − 3.99·23-s − 0.104·24-s + 0.0635·25-s − 0.347·26-s − 1.86·27-s + 6.39·28-s + ⋯
L(s)  = 1  + 0.0587·2-s + 0.182·3-s − 0.996·4-s + 1.00·5-s + 0.0107·6-s − 1.21·7-s − 0.117·8-s − 0.966·9-s + 0.0590·10-s − 0.0657·11-s − 0.181·12-s − 1.15·13-s − 0.0711·14-s + 0.183·15-s + 0.989·16-s − 1.13·17-s − 0.0567·18-s + 0.789·19-s − 1.00·20-s − 0.220·21-s − 0.00386·22-s − 0.833·23-s − 0.0213·24-s + 0.0127·25-s − 0.0680·26-s − 0.358·27-s + 1.20·28-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: $-1$
Motivic weight: \(1\)
Character: $\chi_{503} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 503,\ (\ :1/2),\ -1)\)

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)$
bad503 \( 1 + T \)
good2 \( 1 - 0.0830T + 2T^{2} \)
3 \( 1 - 0.315T + 3T^{2} \)
5 \( 1 - 2.25T + 5T^{2} \)
7 \( 1 + 3.20T + 7T^{2} \)
11 \( 1 + 0.218T + 11T^{2} \)
13 \( 1 + 4.17T + 13T^{2} \)
17 \( 1 + 4.68T + 17T^{2} \)
19 \( 1 - 3.43T + 19T^{2} \)
23 \( 1 + 3.99T + 23T^{2} \)
29 \( 1 + 0.712T + 29T^{2} \)
31 \( 1 - 1.04T + 31T^{2} \)
37 \( 1 - 1.70T + 37T^{2} \)
41 \( 1 - 3.18T + 41T^{2} \)
43 \( 1 + 6.35T + 43T^{2} \)
47 \( 1 + 3.87T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 + 2.63T + 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 - 6.09T + 67T^{2} \)
71 \( 1 - 9.40T + 71T^{2} \)
73 \( 1 + 2.78T + 73T^{2} \)
79 \( 1 - 5.16T + 79T^{2} \)
83 \( 1 - 1.83T + 83T^{2} \)
89 \( 1 + 8.08T + 89T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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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

−9.956274510837681971206349104133, −9.700735021215489979572089006310, −8.951367158355819332615274656248, −7.931231294352630285822202401773, −6.58118395239002092219668518263, −5.76097042951575866711337732612, −4.85735929312955023060199259222, −3.47317555755971725644831080250, −2.38032565242628041145185462801, 0, 2.38032565242628041145185462801, 3.47317555755971725644831080250, 4.85735929312955023060199259222, 5.76097042951575866711337732612, 6.58118395239002092219668518263, 7.931231294352630285822202401773, 8.951367158355819332615274656248, 9.700735021215489979572089006310, 9.956274510837681971206349104133

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