# Properties

 Label 2-245-1.1-c5-0-31 Degree $2$ Conductor $245$ Sign $-1$ Analytic cond. $39.2940$ Root an. cond. $6.26849$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 0.350·2-s − 21.5·3-s − 31.8·4-s − 25·5-s − 7.56·6-s − 22.3·8-s + 223.·9-s − 8.76·10-s − 193.·11-s + 688.·12-s + 691.·13-s + 539.·15-s + 1.01e3·16-s + 649.·17-s + 78.2·18-s + 696.·19-s + 796.·20-s − 67.7·22-s − 2.97e3·23-s + 483.·24-s + 625·25-s + 242.·26-s + 428.·27-s + 6.77e3·29-s + 189.·30-s + 151.·31-s + 1.07e3·32-s + ⋯
 L(s)  = 1 + 0.0619·2-s − 1.38·3-s − 0.996·4-s − 0.447·5-s − 0.0858·6-s − 0.123·8-s + 0.918·9-s − 0.0277·10-s − 0.481·11-s + 1.37·12-s + 1.13·13-s + 0.619·15-s + 0.988·16-s + 0.545·17-s + 0.0569·18-s + 0.442·19-s + 0.445·20-s − 0.0298·22-s − 1.17·23-s + 0.171·24-s + 0.200·25-s + 0.0702·26-s + 0.113·27-s + 1.49·29-s + 0.0383·30-s + 0.0283·31-s + 0.184·32-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: no Arithmetic: yes Character: $\chi_{245} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 245,\ (\ :5/2),\ -1)$$

## Particular Values

 $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + 25T$$
7 $$1$$
good2 $$1 - 0.350T + 32T^{2}$$
3 $$1 + 21.5T + 243T^{2}$$
11 $$1 + 193.T + 1.61e5T^{2}$$
13 $$1 - 691.T + 3.71e5T^{2}$$
17 $$1 - 649.T + 1.41e6T^{2}$$
19 $$1 - 696.T + 2.47e6T^{2}$$
23 $$1 + 2.97e3T + 6.43e6T^{2}$$
29 $$1 - 6.77e3T + 2.05e7T^{2}$$
31 $$1 - 151.T + 2.86e7T^{2}$$
37 $$1 + 1.10e4T + 6.93e7T^{2}$$
41 $$1 - 7.36e3T + 1.15e8T^{2}$$
43 $$1 + 1.92e4T + 1.47e8T^{2}$$
47 $$1 - 1.27e4T + 2.29e8T^{2}$$
53 $$1 - 9.07e3T + 4.18e8T^{2}$$
59 $$1 + 3.69e4T + 7.14e8T^{2}$$
61 $$1 + 2.34e4T + 8.44e8T^{2}$$
67 $$1 - 6.44e4T + 1.35e9T^{2}$$
71 $$1 + 8.31e4T + 1.80e9T^{2}$$
73 $$1 - 2.89e4T + 2.07e9T^{2}$$
79 $$1 - 1.69e3T + 3.07e9T^{2}$$
83 $$1 - 8.03e4T + 3.93e9T^{2}$$
89 $$1 - 1.18e5T + 5.58e9T^{2}$$
97 $$1 + 1.47e5T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$