# Properties

 Label 2-245-1.1-c5-0-12 Degree $2$ Conductor $245$ Sign $1$ Analytic cond. $39.2940$ Root an. cond. $6.26849$ Motivic weight $5$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 2·2-s + 4·3-s − 28·4-s − 25·5-s + 8·6-s − 120·8-s − 227·9-s − 50·10-s − 148·11-s − 112·12-s − 286·13-s − 100·15-s + 656·16-s + 1.67e3·17-s − 454·18-s − 1.06e3·19-s + 700·20-s − 296·22-s + 2.97e3·23-s − 480·24-s + 625·25-s − 572·26-s − 1.88e3·27-s − 3.41e3·29-s − 200·30-s + 2.44e3·31-s + 5.15e3·32-s + ⋯
 L(s)  = 1 + 0.353·2-s + 0.256·3-s − 7/8·4-s − 0.447·5-s + 0.0907·6-s − 0.662·8-s − 0.934·9-s − 0.158·10-s − 0.368·11-s − 0.224·12-s − 0.469·13-s − 0.114·15-s + 0.640·16-s + 1.40·17-s − 0.330·18-s − 0.673·19-s + 0.391·20-s − 0.130·22-s + 1.17·23-s − 0.170·24-s + 1/5·25-s − 0.165·26-s − 0.496·27-s − 0.752·29-s − 0.0405·30-s + 0.457·31-s + 0.889·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$245$$    =    $$5 \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$39.2940$$ Root analytic conductor: $$6.26849$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: $\chi_{245} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 245,\ (\ :5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.362191020$$ $$L(\frac12)$$ $$\approx$$ $$1.362191020$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + p^{2} T$$
7 $$1$$
good2 $$1 - p T + p^{5} T^{2}$$
3 $$1 - 4 T + p^{5} T^{2}$$
11 $$1 + 148 T + p^{5} T^{2}$$
13 $$1 + 22 p T + p^{5} T^{2}$$
17 $$1 - 1678 T + p^{5} T^{2}$$
19 $$1 + 1060 T + p^{5} T^{2}$$
23 $$1 - 2976 T + p^{5} T^{2}$$
29 $$1 + 3410 T + p^{5} T^{2}$$
31 $$1 - 2448 T + p^{5} T^{2}$$
37 $$1 - 182 T + p^{5} T^{2}$$
41 $$1 - 9398 T + p^{5} T^{2}$$
43 $$1 + 1244 T + p^{5} T^{2}$$
47 $$1 - 12088 T + p^{5} T^{2}$$
53 $$1 - 23846 T + p^{5} T^{2}$$
59 $$1 - 20020 T + p^{5} T^{2}$$
61 $$1 + 32302 T + p^{5} T^{2}$$
67 $$1 - 60972 T + p^{5} T^{2}$$
71 $$1 + 32648 T + p^{5} T^{2}$$
73 $$1 - 38774 T + p^{5} T^{2}$$
79 $$1 + 33360 T + p^{5} T^{2}$$
83 $$1 + 16716 T + p^{5} T^{2}$$
89 $$1 + 101370 T + p^{5} T^{2}$$
97 $$1 - 119038 T + p^{5} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$