# Properties

 Label 2-725-145.17-c1-0-31 Degree $2$ Conductor $725$ Sign $0.934 + 0.355i$ Analytic cond. $5.78915$ Root an. cond. $2.40606$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.23·2-s + 3.00·4-s + (−2.23 − 2.23i)7-s + 2.23·8-s + 3·9-s + (4 − 4i)11-s + (4.47 + 4.47i)13-s + (−5.00 − 5.00i)14-s − 0.999·16-s − 4.47·17-s + 6.70·18-s + (2 + 2i)19-s + (8.94 − 8.94i)22-s + (2.23 − 2.23i)23-s + (10.0 + 10.0i)26-s + ⋯
 L(s)  = 1 + 1.58·2-s + 1.50·4-s + (−0.845 − 0.845i)7-s + 0.790·8-s + 9-s + (1.20 − 1.20i)11-s + (1.24 + 1.24i)13-s + (−1.33 − 1.33i)14-s − 0.249·16-s − 1.08·17-s + 1.58·18-s + (0.458 + 0.458i)19-s + (1.90 − 1.90i)22-s + (0.466 − 0.466i)23-s + (1.96 + 1.96i)26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$725$$    =    $$5^{2} \cdot 29$$ Sign: $0.934 + 0.355i$ Analytic conductor: $$5.78915$$ Root analytic conductor: $$2.40606$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{725} (307, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 725,\ (\ :1/2),\ 0.934 + 0.355i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$3.59874 - 0.661037i$$ $$L(\frac12)$$ $$\approx$$ $$3.59874 - 0.661037i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
29 $$1 + (2 + 5i)T$$
good2 $$1 - 2.23T + 2T^{2}$$
3 $$1 - 3T^{2}$$
7 $$1 + (2.23 + 2.23i)T + 7iT^{2}$$
11 $$1 + (-4 + 4i)T - 11iT^{2}$$
13 $$1 + (-4.47 - 4.47i)T + 13iT^{2}$$
17 $$1 + 4.47T + 17T^{2}$$
19 $$1 + (-2 - 2i)T + 19iT^{2}$$
23 $$1 + (-2.23 + 2.23i)T - 23iT^{2}$$
31 $$1 + (4 - 4i)T - 31iT^{2}$$
37 $$1 - 4.47iT - 37T^{2}$$
41 $$1 + (-1 - i)T + 41iT^{2}$$
43 $$1 - 4.47iT - 43T^{2}$$
47 $$1 + 4.47iT - 47T^{2}$$
53 $$1 + (8.94 - 8.94i)T - 53iT^{2}$$
59 $$1 - 59T^{2}$$
61 $$1 + (9 - 9i)T - 61iT^{2}$$
67 $$1 + (2.23 - 2.23i)T - 67iT^{2}$$
71 $$1 + 12iT - 71T^{2}$$
73 $$1 + 4.47T + 73T^{2}$$
79 $$1 + (-2 - 2i)T + 79iT^{2}$$
83 $$1 + (6.70 - 6.70i)T - 83iT^{2}$$
89 $$1 + (-7 - 7i)T + 89iT^{2}$$
97 $$1 + 4.47iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$