# Properties

 Label 2-78-39.32-c1-0-3 Degree $2$ Conductor $78$ Sign $0.714 + 0.699i$ Analytic cond. $0.622833$ Root an. cond. $0.789197$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.965 − 0.258i)2-s + (−0.933 − 1.45i)3-s + (0.866 − 0.499i)4-s + (0.313 + 0.313i)5-s + (−1.27 − 1.16i)6-s + (−0.0745 + 0.278i)7-s + (0.707 − 0.707i)8-s + (−1.25 + 2.72i)9-s + (0.383 + 0.221i)10-s + (0.150 + 0.563i)11-s + (−1.53 − 0.796i)12-s + (−1.79 + 3.12i)13-s + 0.288i·14-s + (0.164 − 0.749i)15-s + (0.500 − 0.866i)16-s + (2.79 + 4.84i)17-s + ⋯
 L(s)  = 1 + (0.683 − 0.183i)2-s + (−0.539 − 0.842i)3-s + (0.433 − 0.249i)4-s + (0.140 + 0.140i)5-s + (−0.522 − 0.476i)6-s + (−0.0281 + 0.105i)7-s + (0.249 − 0.249i)8-s + (−0.418 + 0.908i)9-s + (0.121 + 0.0700i)10-s + (0.0454 + 0.169i)11-s + (−0.444 − 0.229i)12-s + (−0.496 + 0.867i)13-s + 0.0770i·14-s + (0.0424 − 0.193i)15-s + (0.125 − 0.216i)16-s + (0.678 + 1.17i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$78$$    =    $$2 \cdot 3 \cdot 13$$ Sign: $0.714 + 0.699i$ Analytic conductor: $$0.622833$$ Root analytic conductor: $$0.789197$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{78} (71, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 78,\ (\ :1/2),\ 0.714 + 0.699i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.06522 - 0.434410i$$ $$L(\frac12)$$ $$\approx$$ $$1.06522 - 0.434410i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.965 + 0.258i)T$$
3 $$1 + (0.933 + 1.45i)T$$
13 $$1 + (1.79 - 3.12i)T$$
good5 $$1 + (-0.313 - 0.313i)T + 5iT^{2}$$
7 $$1 + (0.0745 - 0.278i)T + (-6.06 - 3.5i)T^{2}$$
11 $$1 + (-0.150 - 0.563i)T + (-9.52 + 5.5i)T^{2}$$
17 $$1 + (-2.79 - 4.84i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (6.79 + 1.81i)T + (16.4 + 9.5i)T^{2}$$
23 $$1 + (-3.32 + 5.76i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (3.57 + 2.06i)T + (14.5 + 25.1i)T^{2}$$
31 $$1 + (1.03 - 1.03i)T - 31iT^{2}$$
37 $$1 + (-6.72 + 1.80i)T + (32.0 - 18.5i)T^{2}$$
41 $$1 + (7.36 - 1.97i)T + (35.5 - 20.5i)T^{2}$$
43 $$1 + (3.26 - 1.88i)T + (21.5 - 37.2i)T^{2}$$
47 $$1 + (-3.71 + 3.71i)T - 47iT^{2}$$
53 $$1 + 3.64iT - 53T^{2}$$
59 $$1 + (3.29 + 0.881i)T + (51.0 + 29.5i)T^{2}$$
61 $$1 + (5.25 + 9.10i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-2.29 - 8.55i)T + (-58.0 + 33.5i)T^{2}$$
71 $$1 + (3.98 - 14.8i)T + (-61.4 - 35.5i)T^{2}$$
73 $$1 + (-3.52 - 3.52i)T + 73iT^{2}$$
79 $$1 - 1.10T + 79T^{2}$$
83 $$1 + (8.23 + 8.23i)T + 83iT^{2}$$
89 $$1 + (-3.64 - 13.6i)T + (-77.0 + 44.5i)T^{2}$$
97 $$1 + (-2.59 - 0.694i)T + (84.0 + 48.5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$