# Properties

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

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.37·2-s + 0.0763·3-s − 0.118·4-s + 1.17·5-s − 0.104·6-s + 0.469·7-s + 2.90·8-s − 2.99·9-s − 1.60·10-s − 5.74·11-s − 0.00902·12-s − 1.85·13-s − 0.643·14-s + 0.0895·15-s − 3.74·16-s + 5.22·17-s + 4.10·18-s + 2.12·19-s − 0.138·20-s + 0.0358·21-s + 7.88·22-s + 0.171·23-s + 0.221·24-s − 3.62·25-s + 2.54·26-s − 0.457·27-s − 0.0554·28-s + ⋯
 L(s)  = 1 − 0.969·2-s + 0.0440·3-s − 0.0591·4-s + 0.524·5-s − 0.0427·6-s + 0.177·7-s + 1.02·8-s − 0.998·9-s − 0.508·10-s − 1.73·11-s − 0.00260·12-s − 0.515·13-s − 0.172·14-s + 0.0231·15-s − 0.937·16-s + 1.26·17-s + 0.968·18-s + 0.487·19-s − 0.0310·20-s + 0.00782·21-s + 1.68·22-s + 0.0358·23-s + 0.0452·24-s − 0.724·25-s + 0.500·26-s − 0.0880·27-s − 0.0104·28-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 + 1.37T + 2T^{2}$$
3 $$1 - 0.0763T + 3T^{2}$$
5 $$1 - 1.17T + 5T^{2}$$
7 $$1 - 0.469T + 7T^{2}$$
11 $$1 + 5.74T + 11T^{2}$$
13 $$1 + 1.85T + 13T^{2}$$
17 $$1 - 5.22T + 17T^{2}$$
19 $$1 - 2.12T + 19T^{2}$$
23 $$1 - 0.171T + 23T^{2}$$
29 $$1 + 6.19T + 29T^{2}$$
31 $$1 + 0.396T + 31T^{2}$$
37 $$1 + 8.17T + 37T^{2}$$
41 $$1 + 12.4T + 41T^{2}$$
43 $$1 + 4.97T + 43T^{2}$$
47 $$1 + 0.521T + 47T^{2}$$
53 $$1 - 8.76T + 53T^{2}$$
59 $$1 - 3.35T + 59T^{2}$$
61 $$1 + 5.38T + 61T^{2}$$
67 $$1 - 8.42T + 67T^{2}$$
71 $$1 - 7.47T + 71T^{2}$$
73 $$1 + 4.60T + 73T^{2}$$
79 $$1 + 17.1T + 79T^{2}$$
83 $$1 - 5.97T + 83T^{2}$$
89 $$1 + 4.25T + 89T^{2}$$
97 $$1 + 2.64T + 97T^{2}$$
show less
$$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

−10.12896078048346964567057109651, −9.781350409519631162744833300989, −8.604328992472127295998784901795, −7.999807757991561427990482986070, −7.21438076680531961872455098010, −5.53258253483804463893796312630, −5.13991594232472990790976595103, −3.29163548459967138294514493509, −1.94242025813742524563653306179, 0, 1.94242025813742524563653306179, 3.29163548459967138294514493509, 5.13991594232472990790976595103, 5.53258253483804463893796312630, 7.21438076680531961872455098010, 7.999807757991561427990482986070, 8.604328992472127295998784901795, 9.781350409519631162744833300989, 10.12896078048346964567057109651