# Properties

 Label 2-245-1.1-c5-0-7 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 − 8·2-s − 3-s + 32·4-s − 25·5-s + 8·6-s − 242·9-s + 200·10-s − 453·11-s − 32·12-s + 969·13-s + 25·15-s − 1.02e3·16-s − 1.63e3·17-s + 1.93e3·18-s + 1.55e3·19-s − 800·20-s + 3.62e3·22-s − 1.65e3·23-s + 625·25-s − 7.75e3·26-s + 485·27-s − 4.98e3·29-s − 200·30-s − 1.19e3·31-s + 8.19e3·32-s + 453·33-s + 1.30e4·34-s + ⋯
 L(s)  = 1 − 1.41·2-s − 0.0641·3-s + 4-s − 0.447·5-s + 0.0907·6-s − 0.995·9-s + 0.632·10-s − 1.12·11-s − 0.0641·12-s + 1.59·13-s + 0.0286·15-s − 16-s − 1.37·17-s + 1.40·18-s + 0.985·19-s − 0.447·20-s + 1.59·22-s − 0.651·23-s + 1/5·25-s − 2.24·26-s + 0.128·27-s − 1.10·29-s − 0.0405·30-s − 0.222·31-s + 1.41·32-s + 0.0724·33-s + 1.94·34-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

 $$L(3)$$ $$\approx$$ $$0.4079287788$$ $$L(\frac12)$$ $$\approx$$ $$0.4079287788$$ $$L(\frac{7}{2})$$ 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^{3} T + p^{5} T^{2}$$
3 $$1 + T + p^{5} T^{2}$$
11 $$1 + 453 T + p^{5} T^{2}$$
13 $$1 - 969 T + p^{5} T^{2}$$
17 $$1 + 1637 T + p^{5} T^{2}$$
19 $$1 - 1550 T + p^{5} T^{2}$$
23 $$1 + 1654 T + p^{5} T^{2}$$
29 $$1 + 4985 T + p^{5} T^{2}$$
31 $$1 + 1192 T + p^{5} T^{2}$$
37 $$1 + 11018 T + p^{5} T^{2}$$
41 $$1 - 1728 T + p^{5} T^{2}$$
43 $$1 + 10814 T + p^{5} T^{2}$$
47 $$1 + 26237 T + p^{5} T^{2}$$
53 $$1 - 25936 T + p^{5} T^{2}$$
59 $$1 - 4580 T + p^{5} T^{2}$$
61 $$1 - 12488 T + p^{5} T^{2}$$
67 $$1 + 15848 T + p^{5} T^{2}$$
71 $$1 - 51792 T + p^{5} T^{2}$$
73 $$1 + 4846 T + p^{5} T^{2}$$
79 $$1 - 62765 T + p^{5} T^{2}$$
83 $$1 - 23644 T + p^{5} T^{2}$$
89 $$1 - 147300 T + p^{5} T^{2}$$
97 $$1 - 8343 T + p^{5} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$