# Properties

 Label 2-405-5.4-c3-0-36 Degree $2$ Conductor $405$ Sign $0.950 + 0.311i$ Analytic cond. $23.8957$ Root an. cond. $4.88833$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Learn more

## Dirichlet series

 L(s)  = 1 + 0.473i·2-s + 7.77·4-s + (−10.6 − 3.48i)5-s + 8.20i·7-s + 7.47i·8-s + (1.64 − 5.03i)10-s − 5.26·11-s − 77.0i·13-s − 3.88·14-s + 58.6·16-s + 88.9i·17-s + 91.7·19-s + (−82.6 − 27.0i)20-s − 2.49i·22-s − 154. i·23-s + ⋯
 L(s)  = 1 + 0.167i·2-s + 0.971·4-s + (−0.950 − 0.311i)5-s + 0.443i·7-s + 0.330i·8-s + (0.0521 − 0.159i)10-s − 0.144·11-s − 1.64i·13-s − 0.0742·14-s + 0.916·16-s + 1.26i·17-s + 1.10·19-s + (−0.923 − 0.302i)20-s − 0.0241i·22-s − 1.40i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$405$$    =    $$3^{4} \cdot 5$$ Sign: $0.950 + 0.311i$ Analytic conductor: $$23.8957$$ Root analytic conductor: $$4.88833$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{405} (244, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 405,\ (\ :3/2),\ 0.950 + 0.311i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.053986354$$ $$L(\frac12)$$ $$\approx$$ $$2.053986354$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1 + (10.6 + 3.48i)T$$
good2 $$1 - 0.473iT - 8T^{2}$$
7 $$1 - 8.20iT - 343T^{2}$$
11 $$1 + 5.26T + 1.33e3T^{2}$$
13 $$1 + 77.0iT - 2.19e3T^{2}$$
17 $$1 - 88.9iT - 4.91e3T^{2}$$
19 $$1 - 91.7T + 6.85e3T^{2}$$
23 $$1 + 154. iT - 1.21e4T^{2}$$
29 $$1 - 168.T + 2.43e4T^{2}$$
31 $$1 - 72.7T + 2.97e4T^{2}$$
37 $$1 + 154. iT - 5.06e4T^{2}$$
41 $$1 + 7.95T + 6.89e4T^{2}$$
43 $$1 - 23.2iT - 7.95e4T^{2}$$
47 $$1 + 301. iT - 1.03e5T^{2}$$
53 $$1 + 344. iT - 1.48e5T^{2}$$
59 $$1 - 251.T + 2.05e5T^{2}$$
61 $$1 - 272.T + 2.26e5T^{2}$$
67 $$1 - 835. iT - 3.00e5T^{2}$$
71 $$1 + 351.T + 3.57e5T^{2}$$
73 $$1 + 522. iT - 3.89e5T^{2}$$
79 $$1 - 700.T + 4.93e5T^{2}$$
83 $$1 - 241. iT - 5.71e5T^{2}$$
89 $$1 - 1.02e3T + 7.04e5T^{2}$$
97 $$1 + 389. iT - 9.12e5T^{2}$$
show more
show less
$$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

−10.74322645379831194242323795430, −10.19770912385192263577053901491, −8.534391717658377293923877711077, −8.080834861520871573899606550287, −7.13616647195230287108638288182, −6.02243979005083481064955965392, −5.09770090519289561175703810635, −3.60111187189801712071253013088, −2.57355310935018226107201399724, −0.828897949149803896013174363248, 1.12023455635093209930695127748, 2.69204902782797785476753285465, 3.71156369714247161329768909417, 4.90091611485964781847661712012, 6.43973273704427732389547734834, 7.21420063353771336007388425586, 7.75325856660422570885664627205, 9.166960718336367041239039114662, 10.09822372354583757866557888041, 11.16214213240224488445914552147