# Properties

 Label 2-2312-1.1-c3-0-40 Degree $2$ Conductor $2312$ Sign $1$ Analytic cond. $136.412$ Root an. cond. $11.6795$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.95·3-s − 4.89·5-s − 4.46·7-s − 18.2·9-s + 60.3·11-s − 56.1·13-s + 14.4·15-s + 134.·19-s + 13.1·21-s + 39.1·23-s − 101.·25-s + 133.·27-s + 113.·29-s − 306.·31-s − 178.·33-s + 21.8·35-s + 61.9·37-s + 165.·39-s − 317.·41-s − 122.·43-s + 89.4·45-s + 303.·47-s − 323.·49-s − 133.·53-s − 295.·55-s − 398.·57-s + 130.·59-s + ⋯
 L(s)  = 1 − 0.568·3-s − 0.437·5-s − 0.240·7-s − 0.677·9-s + 1.65·11-s − 1.19·13-s + 0.248·15-s + 1.62·19-s + 0.136·21-s + 0.355·23-s − 0.808·25-s + 0.953·27-s + 0.728·29-s − 1.77·31-s − 0.940·33-s + 0.105·35-s + 0.275·37-s + 0.681·39-s − 1.20·41-s − 0.435·43-s + 0.296·45-s + 0.941·47-s − 0.941·49-s − 0.346·53-s − 0.724·55-s − 0.925·57-s + 0.288·59-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2312$$    =    $$2^{3} \cdot 17^{2}$$ Sign: $1$ Analytic conductor: $$136.412$$ Root analytic conductor: $$11.6795$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{2312} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 2312,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.096648735$$ $$L(\frac12)$$ $$\approx$$ $$1.096648735$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
17 $$1$$
good3 $$1 + 2.95T + 27T^{2}$$
5 $$1 + 4.89T + 125T^{2}$$
7 $$1 + 4.46T + 343T^{2}$$
11 $$1 - 60.3T + 1.33e3T^{2}$$
13 $$1 + 56.1T + 2.19e3T^{2}$$
19 $$1 - 134.T + 6.85e3T^{2}$$
23 $$1 - 39.1T + 1.21e4T^{2}$$
29 $$1 - 113.T + 2.43e4T^{2}$$
31 $$1 + 306.T + 2.97e4T^{2}$$
37 $$1 - 61.9T + 5.06e4T^{2}$$
41 $$1 + 317.T + 6.89e4T^{2}$$
43 $$1 + 122.T + 7.95e4T^{2}$$
47 $$1 - 303.T + 1.03e5T^{2}$$
53 $$1 + 133.T + 1.48e5T^{2}$$
59 $$1 - 130.T + 2.05e5T^{2}$$
61 $$1 + 772.T + 2.26e5T^{2}$$
67 $$1 - 378.T + 3.00e5T^{2}$$
71 $$1 - 465.T + 3.57e5T^{2}$$
73 $$1 + 664.T + 3.89e5T^{2}$$
79 $$1 + 925.T + 4.93e5T^{2}$$
83 $$1 - 723.T + 5.71e5T^{2}$$
89 $$1 - 889.T + 7.04e5T^{2}$$
97 $$1 + 1.50e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$