Properties

Degree $2$
Conductor $503$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 1.36·2-s − 2.40·3-s − 0.147·4-s + 0.590·5-s − 3.27·6-s + 1.95·7-s − 2.92·8-s + 2.78·9-s + 0.803·10-s − 1.52·11-s + 0.354·12-s − 4.67·13-s + 2.66·14-s − 1.41·15-s − 3.68·16-s − 3.04·17-s + 3.79·18-s + 0.338·19-s − 0.0869·20-s − 4.71·21-s − 2.07·22-s − 7.98·23-s + 7.03·24-s − 4.65·25-s − 6.36·26-s + 0.513·27-s − 0.288·28-s + ⋯
L(s)  = 1  + 0.962·2-s − 1.38·3-s − 0.0736·4-s + 0.263·5-s − 1.33·6-s + 0.740·7-s − 1.03·8-s + 0.928·9-s + 0.254·10-s − 0.460·11-s + 0.102·12-s − 1.29·13-s + 0.712·14-s − 0.366·15-s − 0.920·16-s − 0.739·17-s + 0.893·18-s + 0.0775·19-s − 0.0194·20-s − 1.02·21-s − 0.443·22-s − 1.66·23-s + 1.43·24-s − 0.930·25-s − 1.24·26-s + 0.0988·27-s − 0.0545·28-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 \( 1 - 1.36T + 2T^{2} \)
3 \( 1 + 2.40T + 3T^{2} \)
5 \( 1 - 0.590T + 5T^{2} \)
7 \( 1 - 1.95T + 7T^{2} \)
11 \( 1 + 1.52T + 11T^{2} \)
13 \( 1 + 4.67T + 13T^{2} \)
17 \( 1 + 3.04T + 17T^{2} \)
19 \( 1 - 0.338T + 19T^{2} \)
23 \( 1 + 7.98T + 23T^{2} \)
29 \( 1 - 6.01T + 29T^{2} \)
31 \( 1 + 4.17T + 31T^{2} \)
37 \( 1 + 11.0T + 37T^{2} \)
41 \( 1 + 4.82T + 41T^{2} \)
43 \( 1 - 12.4T + 43T^{2} \)
47 \( 1 - 11.0T + 47T^{2} \)
53 \( 1 + 3.43T + 53T^{2} \)
59 \( 1 - 2.63T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 - 9.66T + 71T^{2} \)
73 \( 1 + 6.33T + 73T^{2} \)
79 \( 1 + 0.667T + 79T^{2} \)
83 \( 1 + 2.26T + 83T^{2} \)
89 \( 1 - 2.21T + 89T^{2} \)
97 \( 1 - 14.2T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.65006738539480287264761411274, −9.908719725802170781487618994965, −8.705603453142486974264896556663, −7.48225901959062073750559493334, −6.35382544184652468611185194022, −5.51334485764104429075438505823, −4.95239572836569212742134842382, −4.08316204472350628528746160400, −2.29259548583577659881946130124, 0, 2.29259548583577659881946130124, 4.08316204472350628528746160400, 4.95239572836569212742134842382, 5.51334485764104429075438505823, 6.35382544184652468611185194022, 7.48225901959062073750559493334, 8.705603453142486974264896556663, 9.908719725802170781487618994965, 10.65006738539480287264761411274

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