# Properties

 Label 2-13-1.1-c3-0-2 Degree $2$ Conductor $13$ Sign $-1$ Analytic cond. $0.767024$ Root an. cond. $0.875799$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

Among degree 2 L-functions of (motivic) weight greater than 1, this is the one of positive rank with the smallest analytic conductor.

## Dirichlet series

 L(s)  = 1 − 5·2-s − 7·3-s + 17·4-s − 7·5-s + 35·6-s − 13·7-s − 45·8-s + 22·9-s + 35·10-s − 26·11-s − 119·12-s + 13·13-s + 65·14-s + 49·15-s + 89·16-s + 77·17-s − 110·18-s − 126·19-s − 119·20-s + 91·21-s + 130·22-s − 96·23-s + 315·24-s − 76·25-s − 65·26-s + 35·27-s − 221·28-s + ⋯
 L(s)  = 1 − 1.76·2-s − 1.34·3-s + 17/8·4-s − 0.626·5-s + 2.38·6-s − 0.701·7-s − 1.98·8-s + 0.814·9-s + 1.10·10-s − 0.712·11-s − 2.86·12-s + 0.277·13-s + 1.24·14-s + 0.843·15-s + 1.39·16-s + 1.09·17-s − 1.44·18-s − 1.52·19-s − 1.33·20-s + 0.945·21-s + 1.25·22-s − 0.870·23-s + 2.67·24-s − 0.607·25-s − 0.490·26-s + 0.249·27-s − 1.49·28-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: yes Arithmetic: yes Character: $\chi_{13} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 13,\ (\ :3/2),\ -1)$$

## Particular Values

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

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad13 $$1 - p T$$
good2 $$1 + 5 T + p^{3} T^{2}$$
3 $$1 + 7 T + p^{3} T^{2}$$
5 $$1 + 7 T + p^{3} T^{2}$$
7 $$1 + 13 T + p^{3} T^{2}$$
11 $$1 + 26 T + p^{3} T^{2}$$
17 $$1 - 77 T + p^{3} T^{2}$$
19 $$1 + 126 T + p^{3} T^{2}$$
23 $$1 + 96 T + p^{3} T^{2}$$
29 $$1 + 82 T + p^{3} T^{2}$$
31 $$1 - 196 T + p^{3} T^{2}$$
37 $$1 + 131 T + p^{3} T^{2}$$
41 $$1 - 336 T + p^{3} T^{2}$$
43 $$1 + 201 T + p^{3} T^{2}$$
47 $$1 + 105 T + p^{3} T^{2}$$
53 $$1 + 432 T + p^{3} T^{2}$$
59 $$1 + 294 T + p^{3} T^{2}$$
61 $$1 + 56 T + p^{3} T^{2}$$
67 $$1 - 478 T + p^{3} T^{2}$$
71 $$1 - 9 T + p^{3} T^{2}$$
73 $$1 - 98 T + p^{3} T^{2}$$
79 $$1 - 1304 T + p^{3} T^{2}$$
83 $$1 + 308 T + p^{3} T^{2}$$
89 $$1 + 1190 T + p^{3} T^{2}$$
97 $$1 - 70 T + p^{3} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$