# Properties

 Label 2-13-1.1-c9-0-7 Degree $2$ Conductor $13$ Sign $-1$ Analytic cond. $6.69546$ Root an. cond. $2.58755$ Motivic weight $9$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 7.35·2-s + 42.6·3-s − 457.·4-s − 1.23e3·5-s + 313.·6-s + 892.·7-s − 7.13e3·8-s − 1.78e4·9-s − 9.09e3·10-s − 2.71e4·11-s − 1.95e4·12-s − 2.85e4·13-s + 6.56e3·14-s − 5.26e4·15-s + 1.81e5·16-s − 3.46e4·17-s − 1.31e5·18-s + 4.28e5·19-s + 5.66e5·20-s + 3.80e4·21-s − 1.99e5·22-s + 2.03e6·23-s − 3.04e5·24-s − 4.24e5·25-s − 2.10e5·26-s − 1.60e6·27-s − 4.08e5·28-s + ⋯
 L(s)  = 1 + 0.325·2-s + 0.303·3-s − 0.894·4-s − 0.884·5-s + 0.0987·6-s + 0.140·7-s − 0.615·8-s − 0.907·9-s − 0.287·10-s − 0.559·11-s − 0.271·12-s − 0.277·13-s + 0.0456·14-s − 0.268·15-s + 0.694·16-s − 0.100·17-s − 0.295·18-s + 0.755·19-s + 0.791·20-s + 0.0426·21-s − 0.181·22-s + 1.51·23-s − 0.187·24-s − 0.217·25-s − 0.0901·26-s − 0.579·27-s − 0.125·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$13$$ Sign: $-1$ Analytic conductor: $$6.69546$$ Root analytic conductor: $$2.58755$$ Motivic weight: $$9$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 13,\ (\ :9/2),\ -1)$$

## Particular Values

 $$L(5)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{11}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad13 $$1 + 2.85e4T$$
good2 $$1 - 7.35T + 512T^{2}$$
3 $$1 - 42.6T + 1.96e4T^{2}$$
5 $$1 + 1.23e3T + 1.95e6T^{2}$$
7 $$1 - 892.T + 4.03e7T^{2}$$
11 $$1 + 2.71e4T + 2.35e9T^{2}$$
17 $$1 + 3.46e4T + 1.18e11T^{2}$$
19 $$1 - 4.28e5T + 3.22e11T^{2}$$
23 $$1 - 2.03e6T + 1.80e12T^{2}$$
29 $$1 + 5.26e6T + 1.45e13T^{2}$$
31 $$1 + 4.15e6T + 2.64e13T^{2}$$
37 $$1 + 7.58e6T + 1.29e14T^{2}$$
41 $$1 + 4.92e6T + 3.27e14T^{2}$$
43 $$1 - 1.71e7T + 5.02e14T^{2}$$
47 $$1 + 2.95e7T + 1.11e15T^{2}$$
53 $$1 + 2.72e7T + 3.29e15T^{2}$$
59 $$1 + 1.13e8T + 8.66e15T^{2}$$
61 $$1 + 3.76e7T + 1.16e16T^{2}$$
67 $$1 - 1.90e8T + 2.72e16T^{2}$$
71 $$1 - 6.87e7T + 4.58e16T^{2}$$
73 $$1 - 3.61e8T + 5.88e16T^{2}$$
79 $$1 + 1.42e8T + 1.19e17T^{2}$$
83 $$1 + 5.80e7T + 1.86e17T^{2}$$
89 $$1 + 8.59e8T + 3.50e17T^{2}$$
97 $$1 - 1.46e9T + 7.60e17T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$