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  − 1.62·2-s − 1.48·3-s + 0.649·4-s − 1.79·5-s + 2.42·6-s + 0.552·7-s + 2.19·8-s − 0.782·9-s + 2.92·10-s + 4.33·11-s − 0.967·12-s + 2.54·13-s − 0.898·14-s + 2.67·15-s − 4.87·16-s + 2.52·17-s + 1.27·18-s − 5.11·19-s − 1.16·20-s − 0.822·21-s − 7.06·22-s − 3.78·23-s − 3.27·24-s − 1.76·25-s − 4.14·26-s + 5.63·27-s + 0.358·28-s + ⋯
L(s)  = 1  − 1.15·2-s − 0.859·3-s + 0.324·4-s − 0.804·5-s + 0.989·6-s + 0.208·7-s + 0.777·8-s − 0.260·9-s + 0.925·10-s + 1.30·11-s − 0.279·12-s + 0.706·13-s − 0.240·14-s + 0.691·15-s − 1.21·16-s + 0.611·17-s + 0.300·18-s − 1.17·19-s − 0.261·20-s − 0.179·21-s − 1.50·22-s − 0.789·23-s − 0.667·24-s − 0.352·25-s − 0.813·26-s + 1.08·27-s + 0.0678·28-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 \( 1 + 1.62T + 2T^{2} \)
3 \( 1 + 1.48T + 3T^{2} \)
5 \( 1 + 1.79T + 5T^{2} \)
7 \( 1 - 0.552T + 7T^{2} \)
11 \( 1 - 4.33T + 11T^{2} \)
13 \( 1 - 2.54T + 13T^{2} \)
17 \( 1 - 2.52T + 17T^{2} \)
19 \( 1 + 5.11T + 19T^{2} \)
23 \( 1 + 3.78T + 23T^{2} \)
29 \( 1 - 0.907T + 29T^{2} \)
31 \( 1 - 0.380T + 31T^{2} \)
37 \( 1 + 5.43T + 37T^{2} \)
41 \( 1 - 5.72T + 41T^{2} \)
43 \( 1 + 9.21T + 43T^{2} \)
47 \( 1 + 8.81T + 47T^{2} \)
53 \( 1 + 6.46T + 53T^{2} \)
59 \( 1 - 3.40T + 59T^{2} \)
61 \( 1 + 1.06T + 61T^{2} \)
67 \( 1 + 0.253T + 67T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 + 16.1T + 73T^{2} \)
79 \( 1 - 7.76T + 79T^{2} \)
83 \( 1 + 16.3T + 83T^{2} \)
89 \( 1 + 3.09T + 89T^{2} \)
97 \( 1 + 4.28T + 97T^{2} \)
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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

−10.50074303401151258078273193847, −9.592089007446108715248903774383, −8.525888499037896985220939319413, −8.124372473121297935502672112506, −6.87902516429313604078927764311, −6.07151067468668771523017632831, −4.69238975365214526165116832123, −3.71132593037249286610528280034, −1.49228221554600116989213035098, 0, 1.49228221554600116989213035098, 3.71132593037249286610528280034, 4.69238975365214526165116832123, 6.07151067468668771523017632831, 6.87902516429313604078927764311, 8.124372473121297935502672112506, 8.525888499037896985220939319413, 9.592089007446108715248903774383, 10.50074303401151258078273193847

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