# Properties

 Label 2-78-13.11-c2-0-3 Degree $2$ Conductor $78$ Sign $-0.679 + 0.733i$ Analytic cond. $2.12534$ Root an. cond. $1.45785$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.366 − 1.36i)2-s + (−0.866 − 1.5i)3-s + (−1.73 − i)4-s + (−1.26 − 1.26i)5-s + (−2.36 + 0.633i)6-s + (−2.5 − 9.33i)7-s + (−2 + 1.99i)8-s + (−1.5 + 2.59i)9-s + (−2.19 + 1.26i)10-s + (9.92 + 2.66i)11-s + 3.46i·12-s + (−11.2 − 6.5i)13-s − 13.6·14-s + (−0.803 + 3i)15-s + (1.99 + 3.46i)16-s + (16.3 + 9.46i)17-s + ⋯
 L(s)  = 1 + (0.183 − 0.683i)2-s + (−0.288 − 0.5i)3-s + (−0.433 − 0.250i)4-s + (−0.253 − 0.253i)5-s + (−0.394 + 0.105i)6-s + (−0.357 − 1.33i)7-s + (−0.250 + 0.249i)8-s + (−0.166 + 0.288i)9-s + (−0.219 + 0.126i)10-s + (0.902 + 0.241i)11-s + 0.288i·12-s + (−0.866 − 0.5i)13-s − 0.975·14-s + (−0.0535 + 0.200i)15-s + (0.124 + 0.216i)16-s + (0.964 + 0.556i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$78$$    =    $$2 \cdot 3 \cdot 13$$ Sign: $-0.679 + 0.733i$ Analytic conductor: $$2.12534$$ Root analytic conductor: $$1.45785$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{78} (37, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 78,\ (\ :1),\ -0.679 + 0.733i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.434034 - 0.993173i$$ $$L(\frac12)$$ $$\approx$$ $$0.434034 - 0.993173i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.366 + 1.36i)T$$
3 $$1 + (0.866 + 1.5i)T$$
13 $$1 + (11.2 + 6.5i)T$$
good5 $$1 + (1.26 + 1.26i)T + 25iT^{2}$$
7 $$1 + (2.5 + 9.33i)T + (-42.4 + 24.5i)T^{2}$$
11 $$1 + (-9.92 - 2.66i)T + (104. + 60.5i)T^{2}$$
17 $$1 + (-16.3 - 9.46i)T + (144.5 + 250. i)T^{2}$$
19 $$1 + (-31.9 + 8.56i)T + (312. - 180.5i)T^{2}$$
23 $$1 + (3.58 - 2.07i)T + (264.5 - 458. i)T^{2}$$
29 $$1 + (-8.66 - 15i)T + (-420.5 + 728. i)T^{2}$$
31 $$1 + (-22.1 - 22.1i)T + 961iT^{2}$$
37 $$1 + (35.7 + 9.58i)T + (1.18e3 + 684.5i)T^{2}$$
41 $$1 + (6.67 - 24.9i)T + (-1.45e3 - 840.5i)T^{2}$$
43 $$1 + (13.2 + 7.66i)T + (924.5 + 1.60e3i)T^{2}$$
47 $$1 + (-7.01 + 7.01i)T - 2.20e3iT^{2}$$
53 $$1 + 61.6T + 2.80e3T^{2}$$
59 $$1 + (17.1 + 63.8i)T + (-3.01e3 + 1.74e3i)T^{2}$$
61 $$1 + (-3.65 + 6.32i)T + (-1.86e3 - 3.22e3i)T^{2}$$
67 $$1 + (10.1 - 37.8i)T + (-3.88e3 - 2.24e3i)T^{2}$$
71 $$1 + (-103. + 27.8i)T + (4.36e3 - 2.52e3i)T^{2}$$
73 $$1 + (67.4 - 67.4i)T - 5.32e3iT^{2}$$
79 $$1 - 11.8T + 6.24e3T^{2}$$
83 $$1 + (-111. - 111. i)T + 6.88e3iT^{2}$$
89 $$1 + (-162. - 43.4i)T + (6.85e3 + 3.96e3i)T^{2}$$
97 $$1 + (-150. + 40.2i)T + (8.14e3 - 4.70e3i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$