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  + 1.17·2-s − 0.791·3-s − 0.624·4-s + 0.178·5-s − 0.928·6-s − 0.0809·7-s − 3.07·8-s − 2.37·9-s + 0.209·10-s − 4.30·11-s + 0.494·12-s + 1.10·13-s − 0.0948·14-s − 0.141·15-s − 2.35·16-s − 3.06·17-s − 2.78·18-s − 2.15·19-s − 0.111·20-s + 0.0640·21-s − 5.04·22-s + 9.15·23-s + 2.43·24-s − 4.96·25-s + 1.29·26-s + 4.25·27-s + 0.0505·28-s + ⋯
L(s)  = 1  + 0.829·2-s − 0.457·3-s − 0.312·4-s + 0.0799·5-s − 0.379·6-s − 0.0305·7-s − 1.08·8-s − 0.791·9-s + 0.0662·10-s − 1.29·11-s + 0.142·12-s + 0.305·13-s − 0.0253·14-s − 0.0365·15-s − 0.589·16-s − 0.743·17-s − 0.655·18-s − 0.495·19-s − 0.0249·20-s + 0.0139·21-s − 1.07·22-s + 1.90·23-s + 0.497·24-s − 0.993·25-s + 0.253·26-s + 0.818·27-s + 0.00955·28-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 \( 1 - 1.17T + 2T^{2} \)
3 \( 1 + 0.791T + 3T^{2} \)
5 \( 1 - 0.178T + 5T^{2} \)
7 \( 1 + 0.0809T + 7T^{2} \)
11 \( 1 + 4.30T + 11T^{2} \)
13 \( 1 - 1.10T + 13T^{2} \)
17 \( 1 + 3.06T + 17T^{2} \)
19 \( 1 + 2.15T + 19T^{2} \)
23 \( 1 - 9.15T + 23T^{2} \)
29 \( 1 + 6.17T + 29T^{2} \)
31 \( 1 + 9.82T + 31T^{2} \)
37 \( 1 - 4.08T + 37T^{2} \)
41 \( 1 - 4.58T + 41T^{2} \)
43 \( 1 - 2.03T + 43T^{2} \)
47 \( 1 + 5.12T + 47T^{2} \)
53 \( 1 - 2.69T + 53T^{2} \)
59 \( 1 - 9.21T + 59T^{2} \)
61 \( 1 + 14.0T + 61T^{2} \)
67 \( 1 - 5.62T + 67T^{2} \)
71 \( 1 + 4.88T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 - 17.4T + 79T^{2} \)
83 \( 1 + 2.48T + 83T^{2} \)
89 \( 1 - 5.57T + 89T^{2} \)
97 \( 1 + 13.9T + 97T^{2} \)
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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

−10.96020347626974776631814317707, −9.493819482079649195599741751773, −8.786952821320320039862228805008, −7.72655579683521662917218541156, −6.43180590257067048186112748000, −5.52154731824581310271598715534, −4.97119951933508755479638605402, −3.69557217551358433867422098067, −2.54896388395685297495362863307, 0, 2.54896388395685297495362863307, 3.69557217551358433867422098067, 4.97119951933508755479638605402, 5.52154731824581310271598715534, 6.43180590257067048186112748000, 7.72655579683521662917218541156, 8.786952821320320039862228805008, 9.493819482079649195599741751773, 10.96020347626974776631814317707

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