# Properties

 Label 2-74-37.6-c2-0-4 Degree $2$ Conductor $74$ Sign $-0.0824 + 0.996i$ Analytic cond. $2.01635$ Root an. cond. $1.41998$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1 − i)2-s − 4i·3-s − 2i·4-s + (3 + 3i)5-s + (−4 − 4i)6-s − 4·7-s + (−2 − 2i)8-s − 7·9-s + 6·10-s − 4i·11-s − 8·12-s + (3 + 3i)13-s + (−4 + 4i)14-s + (12 − 12i)15-s − 4·16-s + (23 + 23i)17-s + ⋯
 L(s)  = 1 + (0.5 − 0.5i)2-s − 1.33i·3-s − 0.5i·4-s + (0.600 + 0.600i)5-s + (−0.666 − 0.666i)6-s − 0.571·7-s + (−0.250 − 0.250i)8-s − 0.777·9-s + 0.600·10-s − 0.363i·11-s − 0.666·12-s + (0.230 + 0.230i)13-s + (−0.285 + 0.285i)14-s + (0.800 − 0.800i)15-s − 0.250·16-s + (1.35 + 1.35i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0824 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0824 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$74$$    =    $$2 \cdot 37$$ Sign: $-0.0824 + 0.996i$ Analytic conductor: $$2.01635$$ Root analytic conductor: $$1.41998$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{74} (43, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 74,\ (\ :1),\ -0.0824 + 0.996i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.10398 - 1.19913i$$ $$L(\frac12)$$ $$\approx$$ $$1.10398 - 1.19913i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1 + i)T$$
37 $$1 - 37iT$$
good3 $$1 + 4iT - 9T^{2}$$
5 $$1 + (-3 - 3i)T + 25iT^{2}$$
7 $$1 + 4T + 49T^{2}$$
11 $$1 + 4iT - 121T^{2}$$
13 $$1 + (-3 - 3i)T + 169iT^{2}$$
17 $$1 + (-23 - 23i)T + 289iT^{2}$$
19 $$1 + (-10 - 10i)T + 361iT^{2}$$
23 $$1 + (10 + 10i)T + 529iT^{2}$$
29 $$1 + (19 - 19i)T - 841iT^{2}$$
31 $$1 + (18 - 18i)T - 961iT^{2}$$
41 $$1 + 74iT - 1.68e3T^{2}$$
43 $$1 + (-42 - 42i)T + 1.84e3iT^{2}$$
47 $$1 + 44T + 2.20e3T^{2}$$
53 $$1 + 80T + 2.80e3T^{2}$$
59 $$1 + (54 + 54i)T + 3.48e3iT^{2}$$
61 $$1 + (3 - 3i)T - 3.72e3iT^{2}$$
67 $$1 - 12iT - 4.48e3T^{2}$$
71 $$1 - 124T + 5.04e3T^{2}$$
73 $$1 - 10iT - 5.32e3T^{2}$$
79 $$1 + (-14 - 14i)T + 6.24e3iT^{2}$$
83 $$1 + 64T + 6.88e3T^{2}$$
89 $$1 + (-17 + 17i)T - 7.92e3iT^{2}$$
97 $$1 + (-129 - 129i)T + 9.40e3iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$