Properties

 Label 2-315-1.1-c3-0-20 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 − 1.56·2-s − 5.56·4-s + 5·5-s − 7·7-s + 21.1·8-s − 7.80·10-s − 10.2·11-s + 34.3·13-s + 10.9·14-s + 11.4·16-s − 82.6·17-s + 90.7·19-s − 27.8·20-s + 16·22-s − 12.1·23-s + 25·25-s − 53.6·26-s + 38.9·28-s − 105.·29-s − 142.·31-s − 187.·32-s + 128.·34-s − 35·35-s + 64.8·37-s − 141.·38-s + 105.·40-s − 195.·41-s + ⋯
 L(s)  = 1 − 0.552·2-s − 0.695·4-s + 0.447·5-s − 0.377·7-s + 0.935·8-s − 0.246·10-s − 0.280·11-s + 0.732·13-s + 0.208·14-s + 0.178·16-s − 1.17·17-s + 1.09·19-s − 0.310·20-s + 0.155·22-s − 0.110·23-s + 0.200·25-s − 0.404·26-s + 0.262·28-s − 0.673·29-s − 0.823·31-s − 1.03·32-s + 0.650·34-s − 0.169·35-s + 0.288·37-s − 0.604·38-s + 0.418·40-s − 0.743·41-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 + 1.56T + 8T^{2}$$
11 $$1 + 10.2T + 1.33e3T^{2}$$
13 $$1 - 34.3T + 2.19e3T^{2}$$
17 $$1 + 82.6T + 4.91e3T^{2}$$
19 $$1 - 90.7T + 6.85e3T^{2}$$
23 $$1 + 12.1T + 1.21e4T^{2}$$
29 $$1 + 105.T + 2.43e4T^{2}$$
31 $$1 + 142.T + 2.97e4T^{2}$$
37 $$1 - 64.8T + 5.06e4T^{2}$$
41 $$1 + 195.T + 6.89e4T^{2}$$
43 $$1 + 319.T + 7.95e4T^{2}$$
47 $$1 + 318.T + 1.03e5T^{2}$$
53 $$1 + 296.T + 1.48e5T^{2}$$
59 $$1 + 284T + 2.05e5T^{2}$$
61 $$1 + 494.T + 2.26e5T^{2}$$
67 $$1 - 549.T + 3.00e5T^{2}$$
71 $$1 + 740.T + 3.57e5T^{2}$$
73 $$1 + 556.T + 3.89e5T^{2}$$
79 $$1 + 376.T + 4.93e5T^{2}$$
83 $$1 - 752.T + 5.71e5T^{2}$$
89 $$1 + 945.T + 7.04e5T^{2}$$
97 $$1 + 180.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$