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  − 0.756·2-s − 3.07·3-s − 1.42·4-s + 0.386·5-s + 2.32·6-s + 0.194·7-s + 2.59·8-s + 6.43·9-s − 0.292·10-s + 2.36·11-s + 4.38·12-s − 1.22·13-s − 0.147·14-s − 1.18·15-s + 0.894·16-s − 5.04·17-s − 4.87·18-s + 4.24·19-s − 0.551·20-s − 0.598·21-s − 1.78·22-s + 1.53·23-s − 7.96·24-s − 4.85·25-s + 0.927·26-s − 10.5·27-s − 0.278·28-s + ⋯
L(s)  = 1  − 0.534·2-s − 1.77·3-s − 0.713·4-s + 0.172·5-s + 0.948·6-s + 0.0736·7-s + 0.916·8-s + 2.14·9-s − 0.0923·10-s + 0.712·11-s + 1.26·12-s − 0.340·13-s − 0.0394·14-s − 0.306·15-s + 0.223·16-s − 1.22·17-s − 1.14·18-s + 0.973·19-s − 0.123·20-s − 0.130·21-s − 0.381·22-s + 0.319·23-s − 1.62·24-s − 0.970·25-s + 0.181·26-s − 2.03·27-s − 0.0525·28-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 \( 1 + 0.756T + 2T^{2} \)
3 \( 1 + 3.07T + 3T^{2} \)
5 \( 1 - 0.386T + 5T^{2} \)
7 \( 1 - 0.194T + 7T^{2} \)
11 \( 1 - 2.36T + 11T^{2} \)
13 \( 1 + 1.22T + 13T^{2} \)
17 \( 1 + 5.04T + 17T^{2} \)
19 \( 1 - 4.24T + 19T^{2} \)
23 \( 1 - 1.53T + 23T^{2} \)
29 \( 1 + 7.31T + 29T^{2} \)
31 \( 1 - 3.33T + 31T^{2} \)
37 \( 1 + 2.17T + 37T^{2} \)
41 \( 1 + 1.04T + 41T^{2} \)
43 \( 1 + 1.53T + 43T^{2} \)
47 \( 1 + 1.08T + 47T^{2} \)
53 \( 1 - 1.55T + 53T^{2} \)
59 \( 1 - 14.8T + 59T^{2} \)
61 \( 1 + 5.45T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + 0.902T + 73T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 - 6.10T + 83T^{2} \)
89 \( 1 - 6.44T + 89T^{2} \)
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

−10.44785986863865743805175267068, −9.728571855113006704074525538469, −8.958250027829183239630253920404, −7.60564557589694798257501436264, −6.74826869901705286779948571615, −5.72375640923234197221972978645, −4.90319260196671514686411914704, −4.02222312527322453501192208442, −1.47409392221607039132248049202, 0, 1.47409392221607039132248049202, 4.02222312527322453501192208442, 4.90319260196671514686411914704, 5.72375640923234197221972978645, 6.74826869901705286779948571615, 7.60564557589694798257501436264, 8.958250027829183239630253920404, 9.728571855113006704074525538469, 10.44785986863865743805175267068

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