# Properties

 Degree 2 Conductor $2 \cdot 31$ Sign $0.989 - 0.141i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.809 + 0.587i)2-s + (1.47 − 0.658i)3-s + (0.309 − 0.951i)4-s + (−0.204 − 0.354i)5-s + (−0.809 + 1.40i)6-s + (0.809 + 0.898i)7-s + (0.309 + 0.951i)8-s + (−0.255 + 0.283i)9-s + (0.373 + 0.166i)10-s + (−2.41 + 0.513i)11-s + (−0.169 − 1.60i)12-s + (0.199 − 1.90i)13-s + (−1.18 − 0.251i)14-s + (−0.535 − 0.388i)15-s + (−0.809 − 0.587i)16-s + (−7.88 − 1.67i)17-s + ⋯
 L(s)  = 1 + (−0.572 + 0.415i)2-s + (0.853 − 0.379i)3-s + (0.154 − 0.475i)4-s + (−0.0914 − 0.158i)5-s + (−0.330 + 0.572i)6-s + (0.305 + 0.339i)7-s + (0.109 + 0.336i)8-s + (−0.0851 + 0.0946i)9-s + (0.118 + 0.0526i)10-s + (−0.727 + 0.154i)11-s + (−0.0488 − 0.464i)12-s + (0.0554 − 0.527i)13-s + (−0.316 − 0.0671i)14-s + (−0.138 − 0.100i)15-s + (−0.202 − 0.146i)16-s + (−1.91 − 0.406i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$62$$    =    $$2 \cdot 31$$ $$\varepsilon$$ = $0.989 - 0.141i$ motivic weight = $$1$$ character : $\chi_{62} (49, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 62,\ (\ :1/2),\ 0.989 - 0.141i)$ $L(1)$ $\approx$ $0.830159 + 0.0589325i$ $L(\frac12)$ $\approx$ $0.830159 + 0.0589325i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;31\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (0.809 - 0.587i)T$$
31 $$1 + (-3.20 - 4.55i)T$$
good3 $$1 + (-1.47 + 0.658i)T + (2.00 - 2.22i)T^{2}$$
5 $$1 + (0.204 + 0.354i)T + (-2.5 + 4.33i)T^{2}$$
7 $$1 + (-0.809 - 0.898i)T + (-0.731 + 6.96i)T^{2}$$
11 $$1 + (2.41 - 0.513i)T + (10.0 - 4.47i)T^{2}$$
13 $$1 + (-0.199 + 1.90i)T + (-12.7 - 2.70i)T^{2}$$
17 $$1 + (7.88 + 1.67i)T + (15.5 + 6.91i)T^{2}$$
19 $$1 + (0.0307 + 0.292i)T + (-18.5 + 3.95i)T^{2}$$
23 $$1 + (-1.96 - 6.05i)T + (-18.6 + 13.5i)T^{2}$$
29 $$1 + (-7.56 + 5.49i)T + (8.96 - 27.5i)T^{2}$$
37 $$1 + (2.71 - 4.70i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + (1.97 + 0.877i)T + (27.4 + 30.4i)T^{2}$$
43 $$1 + (0.558 + 5.31i)T + (-42.0 + 8.94i)T^{2}$$
47 $$1 + (-1.19 - 0.870i)T + (14.5 + 44.6i)T^{2}$$
53 $$1 + (-1.93 + 2.15i)T + (-5.54 - 52.7i)T^{2}$$
59 $$1 + (-8.67 + 3.86i)T + (39.4 - 43.8i)T^{2}$$
61 $$1 - 1.44T + 61T^{2}$$
67 $$1 + (-4.02 - 6.96i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + (-5.51 + 6.12i)T + (-7.42 - 70.6i)T^{2}$$
73 $$1 + (6.70 - 1.42i)T + (66.6 - 29.6i)T^{2}$$
79 $$1 + (5.91 + 1.25i)T + (72.1 + 32.1i)T^{2}$$
83 $$1 + (4.61 + 2.05i)T + (55.5 + 61.6i)T^{2}$$
89 $$1 + (0.362 - 1.11i)T + (-72.0 - 52.3i)T^{2}$$
97 $$1 + (3.44 - 10.6i)T + (-78.4 - 57.0i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}