# Properties

 Label 2-13-1.1-c3-0-1 Degree $2$ Conductor $13$ Sign $1$ Analytic cond. $0.767024$ Root an. cond. $0.875799$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.56·2-s − 3.68·3-s − 1.43·4-s + 0.561·5-s − 9.43·6-s + 18.1·7-s − 24.1·8-s − 13.4·9-s + 1.43·10-s + 64.7·11-s + 5.30·12-s − 13·13-s + 46.5·14-s − 2.06·15-s − 50.4·16-s − 25.5·17-s − 34.3·18-s − 107.·19-s − 0.807·20-s − 66.9·21-s + 165.·22-s + 73.2·23-s + 89.0·24-s − 124.·25-s − 33.3·26-s + 148.·27-s − 26.1·28-s + ⋯
 L(s)  = 1 + 0.905·2-s − 0.709·3-s − 0.179·4-s + 0.0502·5-s − 0.642·6-s + 0.981·7-s − 1.06·8-s − 0.497·9-s + 0.0454·10-s + 1.77·11-s + 0.127·12-s − 0.277·13-s + 0.888·14-s − 0.0356·15-s − 0.787·16-s − 0.364·17-s − 0.450·18-s − 1.30·19-s − 0.00903·20-s − 0.695·21-s + 1.60·22-s + 0.664·23-s + 0.757·24-s − 0.997·25-s − 0.251·26-s + 1.06·27-s − 0.176·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$13$$ Sign: $1$ Analytic conductor: $$0.767024$$ Root analytic conductor: $$0.875799$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 13,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.107518455$$ $$L(\frac12)$$ $$\approx$$ $$1.107518455$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad13 $$1 + 13T$$
good2 $$1 - 2.56T + 8T^{2}$$
3 $$1 + 3.68T + 27T^{2}$$
5 $$1 - 0.561T + 125T^{2}$$
7 $$1 - 18.1T + 343T^{2}$$
11 $$1 - 64.7T + 1.33e3T^{2}$$
17 $$1 + 25.5T + 4.91e3T^{2}$$
19 $$1 + 107.T + 6.85e3T^{2}$$
23 $$1 - 73.2T + 1.21e4T^{2}$$
29 $$1 - 175.T + 2.43e4T^{2}$$
31 $$1 + 113.T + 2.97e4T^{2}$$
37 $$1 - 114.T + 5.06e4T^{2}$$
41 $$1 + 69.6T + 6.89e4T^{2}$$
43 $$1 - 438.T + 7.95e4T^{2}$$
47 $$1 + 31.9T + 1.03e5T^{2}$$
53 $$1 - 2.84T + 1.48e5T^{2}$$
59 $$1 - 71.6T + 2.05e5T^{2}$$
61 $$1 + 920.T + 2.26e5T^{2}$$
67 $$1 + 444.T + 3.00e5T^{2}$$
71 $$1 + 541.T + 3.57e5T^{2}$$
73 $$1 - 764.T + 3.89e5T^{2}$$
79 $$1 + 421.T + 4.93e5T^{2}$$
83 $$1 - 603.T + 5.71e5T^{2}$$
89 $$1 + 1.15e3T + 7.04e5T^{2}$$
97 $$1 - 583.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$