# Properties

 Label 2-74-37.31-c4-0-0 Degree $2$ Conductor $74$ Sign $-0.876 + 0.481i$ Analytic cond. $7.64937$ Root an. cond. $2.76575$ Motivic weight $4$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−2 − 2i)2-s + 13.0i·3-s + 8i·4-s + (−23.5 + 23.5i)5-s + (26.1 − 26.1i)6-s − 19.7·7-s + (16 − 16i)8-s − 90.5·9-s + 94.1·10-s − 152. i·11-s − 104.·12-s + (144. − 144. i)13-s + (39.5 + 39.5i)14-s + (−308. − 308. i)15-s − 64·16-s + (−104. + 104. i)17-s + ⋯
 L(s)  = 1 + (−0.5 − 0.5i)2-s + 1.45i·3-s + 0.5i·4-s + (−0.941 + 0.941i)5-s + (0.727 − 0.727i)6-s − 0.403·7-s + (0.250 − 0.250i)8-s − 1.11·9-s + 0.941·10-s − 1.25i·11-s − 0.727·12-s + (0.853 − 0.853i)13-s + (0.201 + 0.201i)14-s + (−1.37 − 1.37i)15-s − 0.250·16-s + (−0.362 + 0.362i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$74$$    =    $$2 \cdot 37$$ Sign: $-0.876 + 0.481i$ Analytic conductor: $$7.64937$$ Root analytic conductor: $$2.76575$$ Motivic weight: $$4$$ Rational: no Arithmetic: yes Character: $\chi_{74} (31, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 74,\ (\ :2),\ -0.876 + 0.481i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$0.0635284 - 0.247467i$$ $$L(\frac12)$$ $$\approx$$ $$0.0635284 - 0.247467i$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (2 + 2i)T$$
37 $$1 + (1.25e3 - 558. i)T$$
good3 $$1 - 13.0iT - 81T^{2}$$
5 $$1 + (23.5 - 23.5i)T - 625iT^{2}$$
7 $$1 + 19.7T + 2.40e3T^{2}$$
11 $$1 + 152. iT - 1.46e4T^{2}$$
13 $$1 + (-144. + 144. i)T - 2.85e4iT^{2}$$
17 $$1 + (104. - 104. i)T - 8.35e4iT^{2}$$
19 $$1 + (227. - 227. i)T - 1.30e5iT^{2}$$
23 $$1 + (346. - 346. i)T - 2.79e5iT^{2}$$
29 $$1 + (831. + 831. i)T + 7.07e5iT^{2}$$
31 $$1 + (-237. - 237. i)T + 9.23e5iT^{2}$$
41 $$1 - 316. iT - 2.82e6T^{2}$$
43 $$1 + (-2.17e3 + 2.17e3i)T - 3.41e6iT^{2}$$
47 $$1 + 613.T + 4.87e6T^{2}$$
53 $$1 + 2.53e3T + 7.89e6T^{2}$$
59 $$1 + (2.73e3 - 2.73e3i)T - 1.21e7iT^{2}$$
61 $$1 + (-1.63e3 - 1.63e3i)T + 1.38e7iT^{2}$$
67 $$1 - 6.63e3iT - 2.01e7T^{2}$$
71 $$1 + 9.61e3T + 2.54e7T^{2}$$
73 $$1 + 5.98e3iT - 2.83e7T^{2}$$
79 $$1 + (-7.62e3 + 7.62e3i)T - 3.89e7iT^{2}$$
83 $$1 + 2.35e3T + 4.74e7T^{2}$$
89 $$1 + (-4.76e3 - 4.76e3i)T + 6.27e7iT^{2}$$
97 $$1 + (-113. + 113. i)T - 8.85e7iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$