# Properties

 Label 2-390-65.7-c1-0-0 Degree $2$ Conductor $390$ Sign $0.263 - 0.964i$ 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 + (0.866 + 0.5i)2-s + (−0.258 − 0.965i)3-s + (0.499 + 0.866i)4-s + (−1.54 + 1.61i)5-s + (0.258 − 0.965i)6-s + (0.954 + 1.65i)7-s + 0.999i·8-s + (−0.866 + 0.499i)9-s + (−2.14 + 0.625i)10-s + (0.562 + 2.09i)11-s + (0.707 − 0.707i)12-s + (2.33 + 2.75i)13-s + 1.90i·14-s + (1.96 + 1.07i)15-s + (−0.5 + 0.866i)16-s + (−0.597 − 0.160i)17-s + ⋯
 L(s)  = 1 + (0.612 + 0.353i)2-s + (−0.149 − 0.557i)3-s + (0.249 + 0.433i)4-s + (−0.691 + 0.722i)5-s + (0.105 − 0.394i)6-s + (0.360 + 0.625i)7-s + 0.353i·8-s + (−0.288 + 0.166i)9-s + (−0.678 + 0.197i)10-s + (0.169 + 0.633i)11-s + (0.204 − 0.204i)12-s + (0.646 + 0.762i)13-s + 0.510i·14-s + (0.506 + 0.277i)15-s + (−0.125 + 0.216i)16-s + (−0.144 − 0.0388i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$390$$    =    $$2 \cdot 3 \cdot 5 \cdot 13$$ Sign: $0.263 - 0.964i$ Analytic conductor: $$3.11416$$ Root analytic conductor: $$1.76469$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{390} (7, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 390,\ (\ :1/2),\ 0.263 - 0.964i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.30462 + 0.996301i$$ $$L(\frac12)$$ $$\approx$$ $$1.30462 + 0.996301i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.866 - 0.5i)T$$
3 $$1 + (0.258 + 0.965i)T$$
5 $$1 + (1.54 - 1.61i)T$$
13 $$1 + (-2.33 - 2.75i)T$$
good7 $$1 + (-0.954 - 1.65i)T + (-3.5 + 6.06i)T^{2}$$
11 $$1 + (-0.562 - 2.09i)T + (-9.52 + 5.5i)T^{2}$$
17 $$1 + (0.597 + 0.160i)T + (14.7 + 8.5i)T^{2}$$
19 $$1 + (-3.46 - 0.927i)T + (16.4 + 9.5i)T^{2}$$
23 $$1 + (0.653 - 0.175i)T + (19.9 - 11.5i)T^{2}$$
29 $$1 + (2.93 + 1.69i)T + (14.5 + 25.1i)T^{2}$$
31 $$1 + (-0.691 - 0.691i)T + 31iT^{2}$$
37 $$1 + (-1.42 + 2.47i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + (-7.83 + 2.09i)T + (35.5 - 20.5i)T^{2}$$
43 $$1 + (0.901 - 3.36i)T + (-37.2 - 21.5i)T^{2}$$
47 $$1 + 10.8T + 47T^{2}$$
53 $$1 + (1.54 - 1.54i)T - 53iT^{2}$$
59 $$1 + (-3.75 + 14.0i)T + (-51.0 - 29.5i)T^{2}$$
61 $$1 + (2.85 + 4.94i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-5.07 - 2.92i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 + (1.78 - 6.64i)T + (-61.4 - 35.5i)T^{2}$$
73 $$1 + 2.15iT - 73T^{2}$$
79 $$1 + 5.11iT - 79T^{2}$$
83 $$1 + 3.84T + 83T^{2}$$
89 $$1 + (-0.804 + 0.215i)T + (77.0 - 44.5i)T^{2}$$
97 $$1 + (-15.8 + 9.16i)T + (48.5 - 84.0i)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

−11.57509979731527667939656278458, −11.09170719838314436133865603635, −9.629856751999293436273775640072, −8.417744162703998625480410990479, −7.57805630121181302133418787815, −6.76393776315258037454314419809, −5.89064386744791690782818654844, −4.63712317041162301474254717729, −3.46862361668157110424213854146, −2.07385929703355323217538245240, 0.999225838513310593171628839774, 3.23977752423170540651702040217, 4.10888949798702989035699935259, 5.04628447247929584882977555491, 5.99606628984974526688938850938, 7.45001271405898834862997580038, 8.391393504115563886854442066007, 9.382103814131145031663482359188, 10.51124390334064627206470911037, 11.22438044431778019762083290846