# Properties

 Label 2-315-1.1-c3-0-25 Degree $2$ Conductor $315$ Sign $-1$ Analytic cond. $18.5856$ Root an. cond. $4.31110$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.70·2-s − 5.10·4-s + 5·5-s + 7·7-s − 22.2·8-s + 8.50·10-s − 37.4·11-s + 29.0·13-s + 11.9·14-s + 2.89·16-s − 58.4·17-s − 54.5·19-s − 25.5·20-s − 63.6·22-s − 161.·23-s + 25·25-s + 49.3·26-s − 35.7·28-s − 137.·29-s + 154.·31-s + 183.·32-s − 99.4·34-s + 35·35-s − 350.·37-s − 92.8·38-s − 111.·40-s − 353.·41-s + ⋯
 L(s)  = 1 + 0.601·2-s − 0.638·4-s + 0.447·5-s + 0.377·7-s − 0.985·8-s + 0.269·10-s − 1.02·11-s + 0.619·13-s + 0.227·14-s + 0.0452·16-s − 0.833·17-s − 0.659·19-s − 0.285·20-s − 0.616·22-s − 1.46·23-s + 0.200·25-s + 0.372·26-s − 0.241·28-s − 0.880·29-s + 0.896·31-s + 1.01·32-s − 0.501·34-s + 0.169·35-s − 1.55·37-s − 0.396·38-s − 0.440·40-s − 1.34·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$315$$    =    $$3^{2} \cdot 5 \cdot 7$$ Sign: $-1$ Analytic conductor: $$18.5856$$ Root analytic conductor: $$4.31110$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 315,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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 - 7T$$
good2 $$1 - 1.70T + 8T^{2}$$
11 $$1 + 37.4T + 1.33e3T^{2}$$
13 $$1 - 29.0T + 2.19e3T^{2}$$
17 $$1 + 58.4T + 4.91e3T^{2}$$
19 $$1 + 54.5T + 6.85e3T^{2}$$
23 $$1 + 161.T + 1.21e4T^{2}$$
29 $$1 + 137.T + 2.43e4T^{2}$$
31 $$1 - 154.T + 2.97e4T^{2}$$
37 $$1 + 350.T + 5.06e4T^{2}$$
41 $$1 + 353.T + 6.89e4T^{2}$$
43 $$1 + 518.T + 7.95e4T^{2}$$
47 $$1 - 542.T + 1.03e5T^{2}$$
53 $$1 + 305.T + 1.48e5T^{2}$$
59 $$1 + 14.6T + 2.05e5T^{2}$$
61 $$1 + 171.T + 2.26e5T^{2}$$
67 $$1 - 551.T + 3.00e5T^{2}$$
71 $$1 - 120.T + 3.57e5T^{2}$$
73 $$1 - 284.T + 3.89e5T^{2}$$
79 $$1 - 941.T + 4.93e5T^{2}$$
83 $$1 + 377.T + 5.71e5T^{2}$$
89 $$1 - 677.T + 7.04e5T^{2}$$
97 $$1 + 1.22e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$