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  − 1.37·2-s + 0.0763·3-s − 0.118·4-s + 1.17·5-s − 0.104·6-s + 0.469·7-s + 2.90·8-s − 2.99·9-s − 1.60·10-s − 5.74·11-s − 0.00902·12-s − 1.85·13-s − 0.643·14-s + 0.0895·15-s − 3.74·16-s + 5.22·17-s + 4.10·18-s + 2.12·19-s − 0.138·20-s + 0.0358·21-s + 7.88·22-s + 0.171·23-s + 0.221·24-s − 3.62·25-s + 2.54·26-s − 0.457·27-s − 0.0554·28-s + ⋯
L(s)  = 1  − 0.969·2-s + 0.0440·3-s − 0.0591·4-s + 0.524·5-s − 0.0427·6-s + 0.177·7-s + 1.02·8-s − 0.998·9-s − 0.508·10-s − 1.73·11-s − 0.00260·12-s − 0.515·13-s − 0.172·14-s + 0.0231·15-s − 0.937·16-s + 1.26·17-s + 0.968·18-s + 0.487·19-s − 0.0310·20-s + 0.00782·21-s + 1.68·22-s + 0.0358·23-s + 0.0452·24-s − 0.724·25-s + 0.500·26-s − 0.0880·27-s − 0.0104·28-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 \( 1 + 1.37T + 2T^{2} \)
3 \( 1 - 0.0763T + 3T^{2} \)
5 \( 1 - 1.17T + 5T^{2} \)
7 \( 1 - 0.469T + 7T^{2} \)
11 \( 1 + 5.74T + 11T^{2} \)
13 \( 1 + 1.85T + 13T^{2} \)
17 \( 1 - 5.22T + 17T^{2} \)
19 \( 1 - 2.12T + 19T^{2} \)
23 \( 1 - 0.171T + 23T^{2} \)
29 \( 1 + 6.19T + 29T^{2} \)
31 \( 1 + 0.396T + 31T^{2} \)
37 \( 1 + 8.17T + 37T^{2} \)
41 \( 1 + 12.4T + 41T^{2} \)
43 \( 1 + 4.97T + 43T^{2} \)
47 \( 1 + 0.521T + 47T^{2} \)
53 \( 1 - 8.76T + 53T^{2} \)
59 \( 1 - 3.35T + 59T^{2} \)
61 \( 1 + 5.38T + 61T^{2} \)
67 \( 1 - 8.42T + 67T^{2} \)
71 \( 1 - 7.47T + 71T^{2} \)
73 \( 1 + 4.60T + 73T^{2} \)
79 \( 1 + 17.1T + 79T^{2} \)
83 \( 1 - 5.97T + 83T^{2} \)
89 \( 1 + 4.25T + 89T^{2} \)
97 \( 1 + 2.64T + 97T^{2} \)
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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

−10.12896078048346964567057109651, −9.781350409519631162744833300989, −8.604328992472127295998784901795, −7.999807757991561427990482986070, −7.21438076680531961872455098010, −5.53258253483804463893796312630, −5.13991594232472990790976595103, −3.29163548459967138294514493509, −1.94242025813742524563653306179, 0, 1.94242025813742524563653306179, 3.29163548459967138294514493509, 5.13991594232472990790976595103, 5.53258253483804463893796312630, 7.21438076680531961872455098010, 7.999807757991561427990482986070, 8.604328992472127295998784901795, 9.781350409519631162744833300989, 10.12896078048346964567057109651

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