# Properties

 Degree 2 Conductor $2 \cdot 31$ Sign $0.524 + 0.851i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.809 − 0.587i)2-s + (−1.75 − 1.27i)3-s + (0.309 − 0.951i)4-s + 1.89·5-s − 2.17·6-s + (−0.5 + 1.53i)7-s + (−0.309 − 0.951i)8-s + (0.533 + 1.64i)9-s + (1.53 − 1.11i)10-s + (−1.87 + 5.77i)11-s + (−1.75 + 1.27i)12-s + (1.34 + 0.973i)13-s + (0.5 + 1.53i)14-s + (−3.34 − 2.42i)15-s + (−0.809 − 0.587i)16-s + (−0.586 − 1.80i)17-s + ⋯
 L(s)  = 1 + (0.572 − 0.415i)2-s + (−1.01 − 0.737i)3-s + (0.154 − 0.475i)4-s + 0.849·5-s − 0.887·6-s + (−0.188 + 0.581i)7-s + (−0.109 − 0.336i)8-s + (0.177 + 0.547i)9-s + (0.485 − 0.353i)10-s + (−0.565 + 1.74i)11-s + (−0.507 + 0.368i)12-s + (0.371 + 0.270i)13-s + (0.133 + 0.411i)14-s + (−0.862 − 0.626i)15-s + (−0.202 − 0.146i)16-s + (−0.142 − 0.438i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$62$$    =    $$2 \cdot 31$$ $$\varepsilon$$ = $0.524 + 0.851i$ motivic weight = $$1$$ character : $\chi_{62} (47, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 62,\ (\ :1/2),\ 0.524 + 0.851i)$ $L(1)$ $\approx$ $0.832002 - 0.464397i$ $L(\frac12)$ $\approx$ $0.832002 - 0.464397i$ $L(\frac{3}{2})$ 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 + (-5.42 + 1.23i)T$$
good3 $$1 + (1.75 + 1.27i)T + (0.927 + 2.85i)T^{2}$$
5 $$1 - 1.89T + 5T^{2}$$
7 $$1 + (0.5 - 1.53i)T + (-5.66 - 4.11i)T^{2}$$
11 $$1 + (1.87 - 5.77i)T + (-8.89 - 6.46i)T^{2}$$
13 $$1 + (-1.34 - 0.973i)T + (4.01 + 12.3i)T^{2}$$
17 $$1 + (0.586 + 1.80i)T + (-13.7 + 9.99i)T^{2}$$
19 $$1 + (-3.12 + 2.26i)T + (5.87 - 18.0i)T^{2}$$
23 $$1 + (2.88 + 8.87i)T + (-18.6 + 13.5i)T^{2}$$
29 $$1 + (7.89 - 5.73i)T + (8.96 - 27.5i)T^{2}$$
37 $$1 + 0.864T + 37T^{2}$$
41 $$1 + (3.03 - 2.20i)T + (12.6 - 38.9i)T^{2}$$
43 $$1 + (-5.63 + 4.09i)T + (13.2 - 40.8i)T^{2}$$
47 $$1 + (1.5 + 1.08i)T + (14.5 + 44.6i)T^{2}$$
53 $$1 + (0.492 + 1.51i)T + (-42.8 + 31.1i)T^{2}$$
59 $$1 + (1.07 + 0.783i)T + (18.2 + 56.1i)T^{2}$$
61 $$1 - 4.11T + 61T^{2}$$
67 $$1 + 6.86T + 67T^{2}$$
71 $$1 + (-0.694 - 2.13i)T + (-57.4 + 41.7i)T^{2}$$
73 $$1 + (-2.02 + 6.24i)T + (-59.0 - 42.9i)T^{2}$$
79 $$1 + (-1.14 - 3.51i)T + (-63.9 + 46.4i)T^{2}$$
83 $$1 + (-6.35 + 4.61i)T + (25.6 - 78.9i)T^{2}$$
89 $$1 + (-3.50 + 10.7i)T + (-72.0 - 52.3i)T^{2}$$
97 $$1 + (-3.43 + 10.5i)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}