# Properties

 Label 2-115-115.22-c3-0-11 Degree $2$ Conductor $115$ Sign $0.964 + 0.265i$ Analytic cond. $6.78521$ Root an. cond. $2.60484$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−2.66 − 2.66i)2-s + (−2.23 + 2.23i)3-s + 6.20i·4-s + (7.41 − 8.37i)5-s + 11.9·6-s + (−10.9 + 10.9i)7-s + (−4.78 + 4.78i)8-s + 17.0i·9-s + (−42.0 + 2.55i)10-s + 18.9i·11-s + (−13.8 − 13.8i)12-s + (43.9 − 43.9i)13-s + 58.1·14-s + (2.14 + 35.2i)15-s + 75.1·16-s + (−38.7 + 38.7i)17-s + ⋯
 L(s)  = 1 + (−0.942 − 0.942i)2-s + (−0.429 + 0.429i)3-s + 0.775i·4-s + (0.662 − 0.748i)5-s + 0.809·6-s + (−0.588 + 0.588i)7-s + (−0.211 + 0.211i)8-s + 0.630i·9-s + (−1.33 + 0.0808i)10-s + 0.519i·11-s + (−0.333 − 0.333i)12-s + (0.937 − 0.937i)13-s + 1.10·14-s + (0.0368 + 0.606i)15-s + 1.17·16-s + (−0.552 + 0.552i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$115$$    =    $$5 \cdot 23$$ Sign: $0.964 + 0.265i$ Analytic conductor: $$6.78521$$ Root analytic conductor: $$2.60484$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{115} (22, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 115,\ (\ :3/2),\ 0.964 + 0.265i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.838400 - 0.113353i$$ $$L(\frac12)$$ $$\approx$$ $$0.838400 - 0.113353i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + (-7.41 + 8.37i)T$$
23 $$1 + (-109. - 10.7i)T$$
good2 $$1 + (2.66 + 2.66i)T + 8iT^{2}$$
3 $$1 + (2.23 - 2.23i)T - 27iT^{2}$$
7 $$1 + (10.9 - 10.9i)T - 343iT^{2}$$
11 $$1 - 18.9iT - 1.33e3T^{2}$$
13 $$1 + (-43.9 + 43.9i)T - 2.19e3iT^{2}$$
17 $$1 + (38.7 - 38.7i)T - 4.91e3iT^{2}$$
19 $$1 - 123.T + 6.85e3T^{2}$$
29 $$1 - 59.0iT - 2.43e4T^{2}$$
31 $$1 - 342.T + 2.97e4T^{2}$$
37 $$1 + (302. - 302. i)T - 5.06e4iT^{2}$$
41 $$1 - 137.T + 6.89e4T^{2}$$
43 $$1 + (-155. - 155. i)T + 7.95e4iT^{2}$$
47 $$1 + (-234. - 234. i)T + 1.03e5iT^{2}$$
53 $$1 + (342. + 342. i)T + 1.48e5iT^{2}$$
59 $$1 + 87.0iT - 2.05e5T^{2}$$
61 $$1 - 565. iT - 2.26e5T^{2}$$
67 $$1 + (4.05 - 4.05i)T - 3.00e5iT^{2}$$
71 $$1 + 556.T + 3.57e5T^{2}$$
73 $$1 + (-305. + 305. i)T - 3.89e5iT^{2}$$
79 $$1 + 328.T + 4.93e5T^{2}$$
83 $$1 + (-810. - 810. i)T + 5.71e5iT^{2}$$
89 $$1 - 194.T + 7.04e5T^{2}$$
97 $$1 + (-1.06e3 + 1.06e3i)T - 9.12e5iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$