# Properties

 Label 2-390-195.149-c1-0-23 Degree $2$ Conductor $390$ Sign $0.0309 + 0.999i$ Analytic cond. $3.11416$ Root an. cond. $1.76469$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.258 + 0.965i)2-s + (−1.05 + 1.37i)3-s + (−0.866 + 0.499i)4-s + (1.49 − 1.66i)5-s + (−1.59 − 0.667i)6-s + (−4.36 − 1.16i)7-s + (−0.707 − 0.707i)8-s + (−0.758 − 2.90i)9-s + (1.99 + 1.01i)10-s + (−1.26 − 4.71i)11-s + (0.231 − 1.71i)12-s + (−2.81 + 2.25i)13-s − 4.51i·14-s + (0.692 + 3.81i)15-s + (0.500 − 0.866i)16-s + (−2.59 + 1.49i)17-s + ⋯
 L(s)  = 1 + (0.183 + 0.683i)2-s + (−0.611 + 0.791i)3-s + (−0.433 + 0.249i)4-s + (0.669 − 0.742i)5-s + (−0.652 − 0.272i)6-s + (−1.64 − 0.441i)7-s + (−0.249 − 0.249i)8-s + (−0.252 − 0.967i)9-s + (0.629 + 0.321i)10-s + (−0.381 − 1.42i)11-s + (0.0667 − 0.495i)12-s + (−0.780 + 0.624i)13-s − 1.20i·14-s + (0.178 + 0.983i)15-s + (0.125 − 0.216i)16-s + (−0.629 + 0.363i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0309 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0309 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$390$$    =    $$2 \cdot 3 \cdot 5 \cdot 13$$ Sign: $0.0309 + 0.999i$ Analytic conductor: $$3.11416$$ Root analytic conductor: $$1.76469$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{390} (149, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 390,\ (\ :1/2),\ 0.0309 + 0.999i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.233644 - 0.226533i$$ $$L(\frac12)$$ $$\approx$$ $$0.233644 - 0.226533i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.258 - 0.965i)T$$
3 $$1 + (1.05 - 1.37i)T$$
5 $$1 + (-1.49 + 1.66i)T$$
13 $$1 + (2.81 - 2.25i)T$$
good7 $$1 + (4.36 + 1.16i)T + (6.06 + 3.5i)T^{2}$$
11 $$1 + (1.26 + 4.71i)T + (-9.52 + 5.5i)T^{2}$$
17 $$1 + (2.59 - 1.49i)T + (8.5 - 14.7i)T^{2}$$
19 $$1 + (2.54 + 0.681i)T + (16.4 + 9.5i)T^{2}$$
23 $$1 + (-0.563 - 0.325i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 + (-6.65 - 3.84i)T + (14.5 + 25.1i)T^{2}$$
31 $$1 + (4.43 - 4.43i)T - 31iT^{2}$$
37 $$1 + (2.35 + 8.79i)T + (-32.0 + 18.5i)T^{2}$$
41 $$1 + (1.58 - 0.425i)T + (35.5 - 20.5i)T^{2}$$
43 $$1 + (-2.34 - 4.05i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (5.70 + 5.70i)T + 47iT^{2}$$
53 $$1 - 7.06T + 53T^{2}$$
59 $$1 + (3.39 + 0.908i)T + (51.0 + 29.5i)T^{2}$$
61 $$1 + (5.24 + 9.07i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (12.9 - 3.46i)T + (58.0 - 33.5i)T^{2}$$
71 $$1 + (1.11 - 4.17i)T + (-61.4 - 35.5i)T^{2}$$
73 $$1 + (1.09 - 1.09i)T - 73iT^{2}$$
79 $$1 + 0.529T + 79T^{2}$$
83 $$1 + (0.150 - 0.150i)T - 83iT^{2}$$
89 $$1 + (-2.02 - 7.53i)T + (-77.0 + 44.5i)T^{2}$$
97 $$1 + (-3.09 + 11.5i)T + (-84.0 - 48.5i)T^{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

−10.83542806881195064261418119150, −10.09815744367378832850300342934, −9.176061841121714734627287680863, −8.672765065755589522597854038743, −6.88515595897144232843195298129, −6.22033282380624821947445243489, −5.39028209191190052369951475795, −4.31162695678183181295536254315, −3.16379568745668297144421870422, −0.20124356959612591537355705254, 2.19810777267828559113852266764, 2.89669670796148207475171387363, 4.75710319768491544167021489427, 5.93301281789661533786047899346, 6.64943749595406589296073065374, 7.51676177145272388290463035210, 9.137889943352302898548774549151, 10.13754569663056574216310138040, 10.36892262648850742650718294695, 11.76836089049333461862817936077