# Properties

 Label 2-370-185.88-c1-0-14 Degree $2$ Conductor $370$ Sign $-0.737 + 0.675i$ 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 + (0.5 − 0.866i)2-s + (−0.465 − 0.124i)3-s + (−0.499 − 0.866i)4-s + (−2.03 − 0.917i)5-s + (−0.341 + 0.341i)6-s + (4.19 + 1.12i)7-s − 0.999·8-s + (−2.39 − 1.38i)9-s + (−1.81 + 1.30i)10-s − 3.56i·11-s + (0.124 + 0.465i)12-s + (−2.59 − 4.48i)13-s + (3.07 − 3.07i)14-s + (0.835 + 0.682i)15-s + (−0.5 + 0.866i)16-s + (−0.739 − 0.426i)17-s + ⋯
 L(s)  = 1 + (0.353 − 0.612i)2-s + (−0.269 − 0.0720i)3-s + (−0.249 − 0.433i)4-s + (−0.911 − 0.410i)5-s + (−0.139 + 0.139i)6-s + (1.58 + 0.425i)7-s − 0.353·8-s + (−0.798 − 0.461i)9-s + (−0.573 + 0.413i)10-s − 1.07i·11-s + (0.0360 + 0.134i)12-s + (−0.718 − 1.24i)13-s + (0.821 − 0.821i)14-s + (0.215 + 0.176i)15-s + (−0.125 + 0.216i)16-s + (−0.179 − 0.103i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.737 + 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$370$$    =    $$2 \cdot 5 \cdot 37$$ Sign: $-0.737 + 0.675i$ Analytic conductor: $$2.95446$$ Root analytic conductor: $$1.71885$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{370} (273, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 370,\ (\ :1/2),\ -0.737 + 0.675i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.406043 - 1.04513i$$ $$L(\frac12)$$ $$\approx$$ $$0.406043 - 1.04513i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.5 + 0.866i)T$$
5 $$1 + (2.03 + 0.917i)T$$
37 $$1 + (2.05 + 5.72i)T$$
good3 $$1 + (0.465 + 0.124i)T + (2.59 + 1.5i)T^{2}$$
7 $$1 + (-4.19 - 1.12i)T + (6.06 + 3.5i)T^{2}$$
11 $$1 + 3.56iT - 11T^{2}$$
13 $$1 + (2.59 + 4.48i)T + (-6.5 + 11.2i)T^{2}$$
17 $$1 + (0.739 + 0.426i)T + (8.5 + 14.7i)T^{2}$$
19 $$1 + (-0.530 + 1.98i)T + (-16.4 - 9.5i)T^{2}$$
23 $$1 + 5.38T + 23T^{2}$$
29 $$1 + (2.49 - 2.49i)T - 29iT^{2}$$
31 $$1 + (-3.87 - 3.87i)T + 31iT^{2}$$
41 $$1 + (-3.80 + 2.19i)T + (20.5 - 35.5i)T^{2}$$
43 $$1 - 10.1T + 43T^{2}$$
47 $$1 + (-6.14 + 6.14i)T - 47iT^{2}$$
53 $$1 + (-12.8 + 3.44i)T + (45.8 - 26.5i)T^{2}$$
59 $$1 + (6.39 - 1.71i)T + (51.0 - 29.5i)T^{2}$$
61 $$1 + (2.12 - 7.94i)T + (-52.8 - 30.5i)T^{2}$$
67 $$1 + (-1.47 + 5.50i)T + (-58.0 - 33.5i)T^{2}$$
71 $$1 + (0.748 + 1.29i)T + (-35.5 + 61.4i)T^{2}$$
73 $$1 + (-3.61 + 3.61i)T - 73iT^{2}$$
79 $$1 + (0.864 - 3.22i)T + (-68.4 - 39.5i)T^{2}$$
83 $$1 + (15.3 - 4.11i)T + (71.8 - 41.5i)T^{2}$$
89 $$1 + (-2.96 - 11.0i)T + (-77.0 + 44.5i)T^{2}$$
97 $$1 - 8.46iT - 97T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−11.15202729333172987556036745001, −10.65396522799608443832896608867, −8.979869859463356352767696674404, −8.396121994136549469852933563364, −7.49993683150140844709260885184, −5.70134927569128932495701406216, −5.17555046750981673011231786102, −3.94183787967552507313462049769, −2.63975869395839692296595985372, −0.71204308536680789555606182039, 2.25251776731401142563249293995, 4.30249887586096972616749292000, 4.56022002657637052621313086986, 5.93985791139258209613255618706, 7.29161905745355315940658291143, 7.72917278451434603347636065405, 8.618913003476963939383616355103, 10.01238427753312840385122635697, 11.15632564967706447323566247375, 11.69639150423451052316400281098