# Properties

 Label 2-65-65.47-c1-0-4 Degree $2$ Conductor $65$ Sign $-0.506 + 0.862i$ Analytic cond. $0.519027$ Root an. cond. $0.720435$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.31i·2-s + (−0.240 + 0.240i)3-s − 3.36·4-s + (−1.55 − 1.60i)5-s + (0.556 + 0.556i)6-s + 3.95·7-s + 3.16i·8-s + 2.88i·9-s + (−3.71 + 3.60i)10-s + (−0.556 + 0.556i)11-s + (0.808 − 0.808i)12-s + (3.60 − 0.0370i)13-s − 9.16i·14-s + (0.759 + 0.0117i)15-s + 0.593·16-s + (−1.16 + 1.16i)17-s + ⋯
 L(s)  = 1 − 1.63i·2-s + (−0.138 + 0.138i)3-s − 1.68·4-s + (−0.696 − 0.717i)5-s + (0.227 + 0.227i)6-s + 1.49·7-s + 1.11i·8-s + 0.961i·9-s + (−1.17 + 1.14i)10-s + (−0.167 + 0.167i)11-s + (0.233 − 0.233i)12-s + (0.999 − 0.0102i)13-s − 2.45i·14-s + (0.196 + 0.00302i)15-s + 0.148·16-s + (−0.281 + 0.281i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$65$$    =    $$5 \cdot 13$$ Sign: $-0.506 + 0.862i$ Analytic conductor: $$0.519027$$ Root analytic conductor: $$0.720435$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{65} (47, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 65,\ (\ :1/2),\ -0.506 + 0.862i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.409966 - 0.716172i$$ $$L(\frac12)$$ $$\approx$$ $$0.409966 - 0.716172i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + (1.55 + 1.60i)T$$
13 $$1 + (-3.60 + 0.0370i)T$$
good2 $$1 + 2.31iT - 2T^{2}$$
3 $$1 + (0.240 - 0.240i)T - 3iT^{2}$$
7 $$1 - 3.95T + 7T^{2}$$
11 $$1 + (0.556 - 0.556i)T - 11iT^{2}$$
17 $$1 + (1.16 - 1.16i)T - 17iT^{2}$$
19 $$1 + (1.24 - 1.24i)T - 19iT^{2}$$
23 $$1 + (2.80 + 2.80i)T + 23iT^{2}$$
29 $$1 - 3.47iT - 29T^{2}$$
31 $$1 + (2.07 + 2.07i)T + 31iT^{2}$$
37 $$1 + 6.84T + 37T^{2}$$
41 $$1 + (-7.52 - 7.52i)T + 41iT^{2}$$
43 $$1 + (7.03 + 7.03i)T + 43iT^{2}$$
47 $$1 - 9.09T + 47T^{2}$$
53 $$1 + (-0.958 + 0.958i)T - 53iT^{2}$$
59 $$1 + (7.46 + 7.46i)T + 59iT^{2}$$
61 $$1 - 3.68T + 61T^{2}$$
67 $$1 + 3.78iT - 67T^{2}$$
71 $$1 + (3.67 + 3.67i)T + 71iT^{2}$$
73 $$1 + 5.57iT - 73T^{2}$$
79 $$1 - 9.03iT - 79T^{2}$$
83 $$1 - 6.16T + 83T^{2}$$
89 $$1 + (3.51 + 3.51i)T + 89iT^{2}$$
97 $$1 + 9.03iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$