# Properties

 Degree 2 Conductor $13 \cdot 31$ Sign $0.973 - 0.227i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 0.624i·2-s − 2.78·3-s + 1.60·4-s + 0.619i·5-s − 1.73i·6-s − 2.35i·7-s + 2.25i·8-s + 4.74·9-s − 0.387·10-s − 1.21i·11-s − 4.48·12-s + (−0.818 − 3.51i)13-s + 1.46·14-s − 1.72i·15-s + 1.80·16-s + 4.00·17-s + ⋯
 L(s)  = 1 + 0.441i·2-s − 1.60·3-s + 0.804·4-s + 0.277i·5-s − 0.710i·6-s − 0.889i·7-s + 0.797i·8-s + 1.58·9-s − 0.122·10-s − 0.366i·11-s − 1.29·12-s + (−0.227 − 0.973i)13-s + 0.392·14-s − 0.445i·15-s + 0.452·16-s + 0.971·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$403$$    =    $$13 \cdot 31$$ $$\varepsilon$$ = $0.973 - 0.227i$ motivic weight = $$1$$ character : $\chi_{403} (311, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 403,\ (\ :1/2),\ 0.973 - 0.227i)$ $L(1)$ $\approx$ $1.02939 + 0.118448i$ $L(\frac12)$ $\approx$ $1.02939 + 0.118448i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{13,\;31\}$, $$F_p(T)$$ is a polynomial of degree 2. If $p \in \{13,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad13 $$1 + (0.818 + 3.51i)T$$
31 $$1 + iT$$
good2 $$1 - 0.624iT - 2T^{2}$$
3 $$1 + 2.78T + 3T^{2}$$
5 $$1 - 0.619iT - 5T^{2}$$
7 $$1 + 2.35iT - 7T^{2}$$
11 $$1 + 1.21iT - 11T^{2}$$
17 $$1 - 4.00T + 17T^{2}$$
19 $$1 - 4.95iT - 19T^{2}$$
23 $$1 - 2.11T + 23T^{2}$$
29 $$1 - 10.4T + 29T^{2}$$
37 $$1 + 10.5iT - 37T^{2}$$
41 $$1 - 1.88iT - 41T^{2}$$
43 $$1 + 5.95T + 43T^{2}$$
47 $$1 - 12.5iT - 47T^{2}$$
53 $$1 + 2.74T + 53T^{2}$$
59 $$1 + 7.46iT - 59T^{2}$$
61 $$1 + 6.44T + 61T^{2}$$
67 $$1 + 13.2iT - 67T^{2}$$
71 $$1 + 2.71iT - 71T^{2}$$
73 $$1 - 7.81iT - 73T^{2}$$
79 $$1 - 2.11T + 79T^{2}$$
83 $$1 + 11.0iT - 83T^{2}$$
89 $$1 - 2.01iT - 89T^{2}$$
97 $$1 + 5.43iT - 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}