# Properties

 Label 2-2312-1.1-c3-0-13 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 − 9.26·3-s + 16.4·5-s − 34.5·7-s + 58.7·9-s − 7.42·11-s − 42.0·13-s − 151.·15-s − 59.9·19-s + 319.·21-s − 49.4·23-s + 144.·25-s − 294.·27-s − 259.·29-s − 92.2·31-s + 68.7·33-s − 566.·35-s + 207.·37-s + 389.·39-s − 176.·41-s + 19.0·43-s + 964.·45-s + 80.1·47-s + 847.·49-s − 319.·53-s − 121.·55-s + 554.·57-s − 11.0·59-s + ⋯
 L(s)  = 1 − 1.78·3-s + 1.46·5-s − 1.86·7-s + 2.17·9-s − 0.203·11-s − 0.896·13-s − 2.61·15-s − 0.723·19-s + 3.32·21-s − 0.448·23-s + 1.15·25-s − 2.09·27-s − 1.66·29-s − 0.534·31-s + 0.362·33-s − 2.73·35-s + 0.922·37-s + 1.59·39-s − 0.673·41-s + 0.0676·43-s + 3.19·45-s + 0.248·47-s + 2.47·49-s − 0.829·53-s − 0.298·55-s + 1.28·57-s − 0.0244·59-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

 $$L(2)$$ $$\approx$$ $$0.2979774806$$ $$L(\frac12)$$ $$\approx$$ $$0.2979774806$$ $$L(\frac{5}{2})$$ 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 + 9.26T + 27T^{2}$$
5 $$1 - 16.4T + 125T^{2}$$
7 $$1 + 34.5T + 343T^{2}$$
11 $$1 + 7.42T + 1.33e3T^{2}$$
13 $$1 + 42.0T + 2.19e3T^{2}$$
19 $$1 + 59.9T + 6.85e3T^{2}$$
23 $$1 + 49.4T + 1.21e4T^{2}$$
29 $$1 + 259.T + 2.43e4T^{2}$$
31 $$1 + 92.2T + 2.97e4T^{2}$$
37 $$1 - 207.T + 5.06e4T^{2}$$
41 $$1 + 176.T + 6.89e4T^{2}$$
43 $$1 - 19.0T + 7.95e4T^{2}$$
47 $$1 - 80.1T + 1.03e5T^{2}$$
53 $$1 + 319.T + 1.48e5T^{2}$$
59 $$1 + 11.0T + 2.05e5T^{2}$$
61 $$1 + 712.T + 2.26e5T^{2}$$
67 $$1 - 484.T + 3.00e5T^{2}$$
71 $$1 + 443.T + 3.57e5T^{2}$$
73 $$1 - 337.T + 3.89e5T^{2}$$
79 $$1 + 840.T + 4.93e5T^{2}$$
83 $$1 + 456.T + 5.71e5T^{2}$$
89 $$1 + 1.20e3T + 7.04e5T^{2}$$
97 $$1 + 638.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$