# Properties

 Label 2-370-185.117-c1-0-16 Degree $2$ Conductor $370$ Sign $-0.993 + 0.117i$ Analytic cond. $2.95446$ Root an. cond. $1.71885$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + (−1.28 − 1.28i)3-s + 4-s + (1.63 − 1.52i)5-s + (1.28 + 1.28i)6-s + (−3.01 − 3.01i)7-s − 8-s + 0.323i·9-s + (−1.63 + 1.52i)10-s + 2.38i·11-s + (−1.28 − 1.28i)12-s + 1.44·13-s + (3.01 + 3.01i)14-s + (−4.07 − 0.148i)15-s + 16-s − 2.09i·17-s + ⋯
 L(s)  = 1 − 0.707·2-s + (−0.744 − 0.744i)3-s + 0.5·4-s + (0.732 − 0.680i)5-s + (0.526 + 0.526i)6-s + (−1.13 − 1.13i)7-s − 0.353·8-s + 0.107i·9-s + (−0.517 + 0.481i)10-s + 0.720i·11-s + (−0.372 − 0.372i)12-s + 0.401·13-s + (0.805 + 0.805i)14-s + (−1.05 − 0.0384i)15-s + 0.250·16-s − 0.508i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$370$$    =    $$2 \cdot 5 \cdot 37$$ Sign: $-0.993 + 0.117i$ Analytic conductor: $$2.95446$$ Root analytic conductor: $$1.71885$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{370} (117, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 370,\ (\ :1/2),\ -0.993 + 0.117i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.0287543 - 0.487122i$$ $$L(\frac12)$$ $$\approx$$ $$0.0287543 - 0.487122i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
5 $$1 + (-1.63 + 1.52i)T$$
37 $$1 + (1.89 - 5.78i)T$$
good3 $$1 + (1.28 + 1.28i)T + 3iT^{2}$$
7 $$1 + (3.01 + 3.01i)T + 7iT^{2}$$
11 $$1 - 2.38iT - 11T^{2}$$
13 $$1 - 1.44T + 13T^{2}$$
17 $$1 + 2.09iT - 17T^{2}$$
19 $$1 + (2.46 - 2.46i)T - 19iT^{2}$$
23 $$1 + 0.168T + 23T^{2}$$
29 $$1 + (4.74 + 4.74i)T + 29iT^{2}$$
31 $$1 + (3.34 - 3.34i)T - 31iT^{2}$$
41 $$1 + 6.22iT - 41T^{2}$$
43 $$1 + 3.07T + 43T^{2}$$
47 $$1 + (4.67 + 4.67i)T + 47iT^{2}$$
53 $$1 + (2.87 - 2.87i)T - 53iT^{2}$$
59 $$1 + (-6.78 + 6.78i)T - 59iT^{2}$$
61 $$1 + (-6.94 + 6.94i)T - 61iT^{2}$$
67 $$1 + (-7.89 + 7.89i)T - 67iT^{2}$$
71 $$1 - 12.6T + 71T^{2}$$
73 $$1 + (-4.83 - 4.83i)T + 73iT^{2}$$
79 $$1 + (5.31 - 5.31i)T - 79iT^{2}$$
83 $$1 + (-8.43 + 8.43i)T - 83iT^{2}$$
89 $$1 + (11.6 + 11.6i)T + 89iT^{2}$$
97 $$1 - 5.04iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$