# Properties

 Label 2-378-21.20-c1-0-10 Degree $2$ Conductor $378$ Sign $-0.755 + 0.654i$ Analytic cond. $3.01834$ Root an. cond. $1.73733$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − i·2-s − 4-s − 1.73·5-s + (2 − 1.73i)7-s + i·8-s + 1.73i·10-s − 3i·11-s − 3.46i·13-s + (−1.73 − 2i)14-s + 16-s − 6.92·17-s − 1.73i·19-s + 1.73·20-s − 3·22-s − 3i·23-s + ⋯
 L(s)  = 1 − 0.707i·2-s − 0.5·4-s − 0.774·5-s + (0.755 − 0.654i)7-s + 0.353i·8-s + 0.547i·10-s − 0.904i·11-s − 0.960i·13-s + (−0.462 − 0.534i)14-s + 0.250·16-s − 1.68·17-s − 0.397i·19-s + 0.387·20-s − 0.639·22-s − 0.625i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$378$$    =    $$2 \cdot 3^{3} \cdot 7$$ Sign: $-0.755 + 0.654i$ Analytic conductor: $$3.01834$$ Root analytic conductor: $$1.73733$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{378} (377, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 378,\ (\ :1/2),\ -0.755 + 0.654i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.336354 - 0.902178i$$ $$L(\frac12)$$ $$\approx$$ $$0.336354 - 0.902178i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + iT$$
3 $$1$$
7 $$1 + (-2 + 1.73i)T$$
good5 $$1 + 1.73T + 5T^{2}$$
11 $$1 + 3iT - 11T^{2}$$
13 $$1 + 3.46iT - 13T^{2}$$
17 $$1 + 6.92T + 17T^{2}$$
19 $$1 + 1.73iT - 19T^{2}$$
23 $$1 + 3iT - 23T^{2}$$
29 $$1 + 6iT - 29T^{2}$$
31 $$1 - 5.19iT - 31T^{2}$$
37 $$1 - 7T + 37T^{2}$$
41 $$1 - 12.1T + 41T^{2}$$
43 $$1 + 2T + 43T^{2}$$
47 $$1 - 3.46T + 47T^{2}$$
53 $$1 - 12iT - 53T^{2}$$
59 $$1 + 3.46T + 59T^{2}$$
61 $$1 + 6.92iT - 61T^{2}$$
67 $$1 - 2T + 67T^{2}$$
71 $$1 - 3iT - 71T^{2}$$
73 $$1 - 3.46iT - 73T^{2}$$
79 $$1 + 10T + 79T^{2}$$
83 $$1 - 17.3T + 83T^{2}$$
89 $$1 - 5.19T + 89T^{2}$$
97 $$1 + 13.8iT - 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

−11.05352991001634789015774597396, −10.49208491166796264591504321122, −9.135122218821324976242156531872, −8.248020643847419198410929074308, −7.54394187372462970620798312318, −6.12390283099408262641506106817, −4.70629313770613856950238198729, −3.94386026353510527376359751277, −2.58436935133502546713499710304, −0.65742208539132196488344514468, 2.09103980985007047506957425150, 4.08282557117265537752476112023, 4.75680924392216707947067771664, 6.04563459151078904046672920376, 7.13760509092754352747438475199, 7.88175738467019999506348158549, 8.860577689329856263759497452619, 9.555270959883553525954986347738, 11.03691457841884819142619487537, 11.65829347179685834056416686401