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  + 2.15·2-s − 1.51·3-s + 2.65·4-s − 2.23·5-s − 3.25·6-s − 3.60·7-s + 1.41·8-s − 0.717·9-s − 4.82·10-s − 1.50·11-s − 4.01·12-s + 0.00459·13-s − 7.77·14-s + 3.37·15-s − 2.25·16-s + 1.11·17-s − 1.54·18-s + 2.49·19-s − 5.93·20-s + 5.44·21-s − 3.24·22-s + 1.72·23-s − 2.13·24-s − 0.00724·25-s + 0.00991·26-s + 5.61·27-s − 9.57·28-s + ⋯
L(s)  = 1  + 1.52·2-s − 0.872·3-s + 1.32·4-s − 0.999·5-s − 1.33·6-s − 1.36·7-s + 0.500·8-s − 0.239·9-s − 1.52·10-s − 0.453·11-s − 1.15·12-s + 0.00127·13-s − 2.07·14-s + 0.871·15-s − 0.564·16-s + 0.271·17-s − 0.365·18-s + 0.573·19-s − 1.32·20-s + 1.18·21-s − 0.691·22-s + 0.359·23-s − 0.436·24-s − 0.00144·25-s + 0.00194·26-s + 1.08·27-s − 1.80·28-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 \( 1 - 2.15T + 2T^{2} \)
3 \( 1 + 1.51T + 3T^{2} \)
5 \( 1 + 2.23T + 5T^{2} \)
7 \( 1 + 3.60T + 7T^{2} \)
11 \( 1 + 1.50T + 11T^{2} \)
13 \( 1 - 0.00459T + 13T^{2} \)
17 \( 1 - 1.11T + 17T^{2} \)
19 \( 1 - 2.49T + 19T^{2} \)
23 \( 1 - 1.72T + 23T^{2} \)
29 \( 1 + 0.572T + 29T^{2} \)
31 \( 1 - 3.25T + 31T^{2} \)
37 \( 1 + 6.13T + 37T^{2} \)
41 \( 1 + 5.89T + 41T^{2} \)
43 \( 1 + 7.66T + 43T^{2} \)
47 \( 1 - 6.44T + 47T^{2} \)
53 \( 1 + 7.82T + 53T^{2} \)
59 \( 1 - 0.253T + 59T^{2} \)
61 \( 1 - 7.08T + 61T^{2} \)
67 \( 1 - 6.16T + 67T^{2} \)
71 \( 1 + 9.32T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 + 1.99T + 79T^{2} \)
83 \( 1 - 6.91T + 83T^{2} \)
89 \( 1 - 3.37T + 89T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−10.88755733215853625200847277800, −9.892496038490913202254136188722, −8.591826703320284408788389356768, −7.25547444473243390696457949041, −6.47242816868127899244487356049, −5.64008773383229354511052733685, −4.81882432559567811103129986471, −3.63041197872983686430071714236, −2.95781778542714237960065650684, 0, 2.95781778542714237960065650684, 3.63041197872983686430071714236, 4.81882432559567811103129986471, 5.64008773383229354511052733685, 6.47242816868127899244487356049, 7.25547444473243390696457949041, 8.591826703320284408788389356768, 9.892496038490913202254136188722, 10.88755733215853625200847277800

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