Properties

 Label 2-315-35.34-c4-0-38 Degree $2$ Conductor $315$ Sign $1$ Analytic cond. $32.5615$ Root an. cond. $5.70627$ Motivic weight $4$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

Origins

Dirichlet series

 L(s)  = 1 + 16·4-s − 25·5-s + 49·7-s + 73·11-s + 23·13-s + 256·16-s − 263·17-s − 400·20-s + 625·25-s + 784·28-s + 1.15e3·29-s − 1.22e3·35-s + 1.16e3·44-s + 3.45e3·47-s + 2.40e3·49-s + 368·52-s − 1.82e3·55-s + 4.09e3·64-s − 575·65-s − 4.20e3·68-s + 1.00e4·71-s − 9.50e3·73-s + 3.57e3·77-s + 1.21e4·79-s − 6.40e3·80-s + 6.38e3·83-s + 6.57e3·85-s + ⋯
 L(s)  = 1 + 4-s − 5-s + 7-s + 0.603·11-s + 0.136·13-s + 16-s − 0.910·17-s − 20-s + 25-s + 28-s + 1.37·29-s − 35-s + 0.603·44-s + 1.56·47-s + 49-s + 0.136·52-s − 0.603·55-s + 64-s − 0.136·65-s − 0.910·68-s + 1.99·71-s − 1.78·73-s + 0.603·77-s + 1.94·79-s − 80-s + 0.926·83-s + 0.910·85-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+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: $$32.5615$$ Root analytic conductor: $$5.70627$$ Motivic weight: $$4$$ Rational: yes Arithmetic: yes Character: $\chi_{315} (244, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 315,\ (\ :2),\ 1)$$

Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$2.529293729$$ $$L(\frac12)$$ $$\approx$$ $$2.529293729$$ $$L(3)$$ 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 + p^{2} T$$
7 $$1 - p^{2} T$$
good2 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
11 $$1 - 73 T + p^{4} T^{2}$$
13 $$1 - 23 T + p^{4} T^{2}$$
17 $$1 + 263 T + p^{4} T^{2}$$
19 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
23 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
29 $$1 - 1153 T + p^{4} T^{2}$$
31 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
37 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
41 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
43 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
47 $$1 - 3457 T + p^{4} T^{2}$$
53 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
59 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
61 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
67 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
71 $$1 - 10078 T + p^{4} T^{2}$$
73 $$1 + 9502 T + p^{4} T^{2}$$
79 $$1 - 12167 T + p^{4} T^{2}$$
83 $$1 - 6382 T + p^{4} T^{2}$$
89 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
97 $$1 - 3383 T + p^{4} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$