# Properties

 Degree 2 Conductor $13 \cdot 31$ Sign $0.255 - 0.966i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.49·2-s + (−1.44 − 2.49i)3-s + 4.21·4-s + (0.778 − 1.34i)5-s + (3.59 + 6.22i)6-s + (−0.277 − 0.481i)7-s − 5.52·8-s + (−2.65 + 4.59i)9-s + (−1.94 + 3.36i)10-s + (−1.79 + 3.11i)11-s + (−6.07 − 10.5i)12-s + (−0.5 + 0.866i)13-s + (0.692 + 1.19i)14-s − 4.48·15-s + 5.34·16-s + (−1.01 − 1.75i)17-s + ⋯
 L(s)  = 1 − 1.76·2-s + (−0.831 − 1.44i)3-s + 2.10·4-s + (0.348 − 0.603i)5-s + (1.46 + 2.53i)6-s + (−0.105 − 0.181i)7-s − 1.95·8-s + (−0.883 + 1.53i)9-s + (−0.613 + 1.06i)10-s + (−0.541 + 0.937i)11-s + (−1.75 − 3.03i)12-s + (−0.138 + 0.240i)13-s + (0.185 + 0.320i)14-s − 1.15·15-s + 1.33·16-s + (−0.245 − 0.425i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$403$$    =    $$13 \cdot 31$$ $$\varepsilon$$ = $0.255 - 0.966i$ motivic weight = $$1$$ character : $\chi_{403} (118, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 403,\ (\ :1/2),\ 0.255 - 0.966i)$$ $$L(1)$$ $$\approx$$ $$0.0583378 + 0.0449459i$$ $$L(\frac12)$$ $$\approx$$ $$0.0583378 + 0.0449459i$$ $$L(\frac{3}{2})$$ 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 + (0.5 - 0.866i)T$$
31 $$1 + (-2.10 - 5.15i)T$$
good2 $$1 + 2.49T + 2T^{2}$$
3 $$1 + (1.44 + 2.49i)T + (-1.5 + 2.59i)T^{2}$$
5 $$1 + (-0.778 + 1.34i)T + (-2.5 - 4.33i)T^{2}$$
7 $$1 + (0.277 + 0.481i)T + (-3.5 + 6.06i)T^{2}$$
11 $$1 + (1.79 - 3.11i)T + (-5.5 - 9.52i)T^{2}$$
17 $$1 + (1.01 + 1.75i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (0.578 + 1.00i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + 8.57T + 23T^{2}$$
29 $$1 + 4.00T + 29T^{2}$$
37 $$1 + (-4.06 - 7.04i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + (5.94 - 10.2i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (1.78 + 3.09i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 - 2.60T + 47T^{2}$$
53 $$1 + (-4.10 + 7.11i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (2.11 + 3.66i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 - 0.963T + 61T^{2}$$
67 $$1 + (2.74 - 4.74i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + (1.95 - 3.38i)T + (-35.5 - 61.4i)T^{2}$$
73 $$1 + (-2.90 + 5.03i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (3.38 + 5.86i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (0.967 - 1.67i)T + (-41.5 - 71.8i)T^{2}$$
89 $$1 - 7.41T + 89T^{2}$$
97 $$1 + 10.7T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}