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  − 2.58·2-s − 1.85·3-s + 4.69·4-s + 1.44·5-s + 4.80·6-s − 1.96·7-s − 6.96·8-s + 0.453·9-s − 3.73·10-s + 2.85·11-s − 8.71·12-s − 3.84·13-s + 5.08·14-s − 2.68·15-s + 8.62·16-s + 1.30·17-s − 1.17·18-s + 3.53·19-s + 6.76·20-s + 3.65·21-s − 7.38·22-s + 4.20·23-s + 12.9·24-s − 2.91·25-s + 9.94·26-s + 4.73·27-s − 9.21·28-s + ⋯
L(s)  = 1  − 1.82·2-s − 1.07·3-s + 2.34·4-s + 0.645·5-s + 1.96·6-s − 0.742·7-s − 2.46·8-s + 0.151·9-s − 1.18·10-s + 0.861·11-s − 2.51·12-s − 1.06·13-s + 1.35·14-s − 0.692·15-s + 2.15·16-s + 0.316·17-s − 0.276·18-s + 0.810·19-s + 1.51·20-s + 0.796·21-s − 1.57·22-s + 0.877·23-s + 2.64·24-s − 0.583·25-s + 1.95·26-s + 0.910·27-s − 1.74·28-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 \( 1 + 2.58T + 2T^{2} \)
3 \( 1 + 1.85T + 3T^{2} \)
5 \( 1 - 1.44T + 5T^{2} \)
7 \( 1 + 1.96T + 7T^{2} \)
11 \( 1 - 2.85T + 11T^{2} \)
13 \( 1 + 3.84T + 13T^{2} \)
17 \( 1 - 1.30T + 17T^{2} \)
19 \( 1 - 3.53T + 19T^{2} \)
23 \( 1 - 4.20T + 23T^{2} \)
29 \( 1 + 1.10T + 29T^{2} \)
31 \( 1 + 3.53T + 31T^{2} \)
37 \( 1 + 6.61T + 37T^{2} \)
41 \( 1 - 0.671T + 41T^{2} \)
43 \( 1 + 4.32T + 43T^{2} \)
47 \( 1 + 0.378T + 47T^{2} \)
53 \( 1 + 7.13T + 53T^{2} \)
59 \( 1 + 13.8T + 59T^{2} \)
61 \( 1 + 6.51T + 61T^{2} \)
67 \( 1 + 7.55T + 67T^{2} \)
71 \( 1 - 0.0744T + 71T^{2} \)
73 \( 1 - 3.29T + 73T^{2} \)
79 \( 1 - 8.57T + 79T^{2} \)
83 \( 1 - 14.8T + 83T^{2} \)
89 \( 1 + 17.5T + 89T^{2} \)
97 \( 1 + 8.01T + 97T^{2} \)
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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

−10.30531193912345880504335877550, −9.514977850268917493838188281342, −9.132329017497648916217430483227, −7.76104253178841227931237929292, −6.84918695592696320815753924977, −6.22873516348755126556100044465, −5.21299582766881042101893983704, −3.04430776434398889730145649122, −1.52148486580040692606158735436, 0, 1.52148486580040692606158735436, 3.04430776434398889730145649122, 5.21299582766881042101893983704, 6.22873516348755126556100044465, 6.84918695592696320815753924977, 7.76104253178841227931237929292, 9.132329017497648916217430483227, 9.514977850268917493838188281342, 10.30531193912345880504335877550

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