# Properties

 Label 2-370-185.68-c1-0-2 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: $\chi_{370} (253, \cdot )$ 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}$$
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$$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

−11.29573403388648904768073967404, −10.93440365544469145184947695348, −9.937847353555167596249882138372, −9.349152029015375479298843832931, −8.436887787965877128561640752249, −6.86059249979427211175617853435, −6.00909457227823162922520337482, −5.46584747974478922396777299584, −3.53985366745794822970880741945, −2.31616812252778997381831312401, 0.43459862122650478939976492554, 1.78589529398794047180456442993, 3.74699968918432392976650693774, 5.36693285188112948028517390926, 6.51447615498759902722389256284, 6.89367623060756308030117597941, 8.087816504146326162100051505116, 9.391317234961806335481965446352, 9.903262935357367270674746772401, 10.81499756231601028421522873400