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  − 2.03·2-s + 1.78·3-s + 2.15·4-s − 0.701·5-s − 3.64·6-s − 2.02·7-s − 0.325·8-s + 0.187·9-s + 1.42·10-s − 0.626·11-s + 3.85·12-s − 2.93·13-s + 4.13·14-s − 1.25·15-s − 3.65·16-s − 2.71·17-s − 0.382·18-s − 1.11·19-s − 1.51·20-s − 3.62·21-s + 1.27·22-s − 0.412·23-s − 0.580·24-s − 4.50·25-s + 5.99·26-s − 5.02·27-s − 4.38·28-s + ⋯
L(s)  = 1  − 1.44·2-s + 1.03·3-s + 1.07·4-s − 0.313·5-s − 1.48·6-s − 0.767·7-s − 0.114·8-s + 0.0624·9-s + 0.452·10-s − 0.189·11-s + 1.11·12-s − 0.814·13-s + 1.10·14-s − 0.323·15-s − 0.913·16-s − 0.657·17-s − 0.0900·18-s − 0.254·19-s − 0.338·20-s − 0.790·21-s + 0.272·22-s − 0.0859·23-s − 0.118·24-s − 0.901·25-s + 1.17·26-s − 0.966·27-s − 0.828·28-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 \( 1 + 2.03T + 2T^{2} \)
3 \( 1 - 1.78T + 3T^{2} \)
5 \( 1 + 0.701T + 5T^{2} \)
7 \( 1 + 2.02T + 7T^{2} \)
11 \( 1 + 0.626T + 11T^{2} \)
13 \( 1 + 2.93T + 13T^{2} \)
17 \( 1 + 2.71T + 17T^{2} \)
19 \( 1 + 1.11T + 19T^{2} \)
23 \( 1 + 0.412T + 23T^{2} \)
29 \( 1 - 6.46T + 29T^{2} \)
31 \( 1 + 4.14T + 31T^{2} \)
37 \( 1 - 2.98T + 37T^{2} \)
41 \( 1 + 0.135T + 41T^{2} \)
43 \( 1 + 0.861T + 43T^{2} \)
47 \( 1 - 1.67T + 47T^{2} \)
53 \( 1 + 8.51T + 53T^{2} \)
59 \( 1 + 0.406T + 59T^{2} \)
61 \( 1 + 8.53T + 61T^{2} \)
67 \( 1 + 13.0T + 67T^{2} \)
71 \( 1 - 2.78T + 71T^{2} \)
73 \( 1 + 2.11T + 73T^{2} \)
79 \( 1 + 0.317T + 79T^{2} \)
83 \( 1 + 5.05T + 83T^{2} \)
89 \( 1 - 16.4T + 89T^{2} \)
97 \( 1 - 7.68T + 97T^{2} \)
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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

−10.09358493493378490230383544202, −9.430095301408504994635374848383, −8.790510999214242274470793903259, −7.964357785666527326501415024540, −7.33198303963524491786428404816, −6.24035767159084886716798266145, −4.51376071483096784551376712580, −3.12983915588039610610705239119, −2.07683284652257114995983312896, 0, 2.07683284652257114995983312896, 3.12983915588039610610705239119, 4.51376071483096784551376712580, 6.24035767159084886716798266145, 7.33198303963524491786428404816, 7.964357785666527326501415024540, 8.790510999214242274470793903259, 9.430095301408504994635374848383, 10.09358493493378490230383544202

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