# Properties

 Label 2-2205-1.1-c3-0-87 Degree $2$ Conductor $2205$ Sign $1$ Analytic cond. $130.099$ Root an. cond. $11.4061$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.644·2-s − 7.58·4-s + 5·5-s + 10.0·8-s − 3.22·10-s + 47.7·11-s + 57.2·13-s + 54.1·16-s + 36.9·17-s + 30.7·19-s − 37.9·20-s − 30.7·22-s − 53.1·23-s + 25·25-s − 36.8·26-s + 195.·29-s + 257.·31-s − 115.·32-s − 23.8·34-s + 346.·37-s − 19.8·38-s + 50.2·40-s − 267.·41-s − 176.·43-s − 361.·44-s + 34.2·46-s + 311.·47-s + ⋯
 L(s)  = 1 − 0.227·2-s − 0.948·4-s + 0.447·5-s + 0.443·8-s − 0.101·10-s + 1.30·11-s + 1.22·13-s + 0.846·16-s + 0.527·17-s + 0.371·19-s − 0.423·20-s − 0.298·22-s − 0.481·23-s + 0.200·25-s − 0.278·26-s + 1.25·29-s + 1.49·31-s − 0.637·32-s − 0.120·34-s + 1.53·37-s − 0.0846·38-s + 0.198·40-s − 1.01·41-s − 0.627·43-s − 1.23·44-s + 0.109·46-s + 0.967·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2205$$    =    $$3^{2} \cdot 5 \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$130.099$$ Root analytic conductor: $$11.4061$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{2205} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 2205,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.370178927$$ $$L(\frac12)$$ $$\approx$$ $$2.370178927$$ $$L(\frac{5}{2})$$ 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 - 5T$$
7 $$1$$
good2 $$1 + 0.644T + 8T^{2}$$
11 $$1 - 47.7T + 1.33e3T^{2}$$
13 $$1 - 57.2T + 2.19e3T^{2}$$
17 $$1 - 36.9T + 4.91e3T^{2}$$
19 $$1 - 30.7T + 6.85e3T^{2}$$
23 $$1 + 53.1T + 1.21e4T^{2}$$
29 $$1 - 195.T + 2.43e4T^{2}$$
31 $$1 - 257.T + 2.97e4T^{2}$$
37 $$1 - 346.T + 5.06e4T^{2}$$
41 $$1 + 267.T + 6.89e4T^{2}$$
43 $$1 + 176.T + 7.95e4T^{2}$$
47 $$1 - 311.T + 1.03e5T^{2}$$
53 $$1 - 492.T + 1.48e5T^{2}$$
59 $$1 + 98.7T + 2.05e5T^{2}$$
61 $$1 - 82.1T + 2.26e5T^{2}$$
67 $$1 - 654.T + 3.00e5T^{2}$$
71 $$1 + 779.T + 3.57e5T^{2}$$
73 $$1 + 829.T + 3.89e5T^{2}$$
79 $$1 + 769.T + 4.93e5T^{2}$$
83 $$1 + 613.T + 5.71e5T^{2}$$
89 $$1 - 457.T + 7.04e5T^{2}$$
97 $$1 + 1.41e3T + 9.12e5T^{2}$$
show less
$$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

−8.472910149659328707598583390885, −8.424601609982491503104505682829, −7.13441248857988855035043823482, −6.24465937673423538220729028828, −5.65742573053319076506289800745, −4.53495372954292475609337195101, −3.94301767491994167299432621593, −2.95120602018056173510434217557, −1.40934051512823791689215379975, −0.848489568283673992685265416503, 0.848489568283673992685265416503, 1.40934051512823791689215379975, 2.95120602018056173510434217557, 3.94301767491994167299432621593, 4.53495372954292475609337195101, 5.65742573053319076506289800745, 6.24465937673423538220729028828, 7.13441248857988855035043823482, 8.424601609982491503104505682829, 8.472910149659328707598583390885