# Properties

 Label 2-91e2-1.1-c1-0-464 Degree $2$ Conductor $8281$ Sign $-1$ Analytic cond. $66.1241$ Root an. cond. $8.13167$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s + 1.41·3-s − 4-s + 4.09·5-s + 1.41·6-s − 3·8-s − 0.999·9-s + 4.09·10-s − 3.79·11-s − 1.41·12-s + 5.79·15-s − 16-s − 1.26·17-s − 0.999·18-s − 2.82·19-s − 4.09·20-s − 3.79·22-s − 7.79·23-s − 4.24·24-s + 11.7·25-s − 5.65·27-s + 0.795·29-s + 5.79·30-s − 1.41·31-s + 5·32-s − 5.36·33-s − 1.26·34-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.816·3-s − 0.5·4-s + 1.83·5-s + 0.577·6-s − 1.06·8-s − 0.333·9-s + 1.29·10-s − 1.14·11-s − 0.408·12-s + 1.49·15-s − 0.250·16-s − 0.307·17-s − 0.235·18-s − 0.648·19-s − 0.916·20-s − 0.809·22-s − 1.62·23-s − 0.866·24-s + 2.35·25-s − 1.08·27-s + 0.147·29-s + 1.05·30-s − 0.254·31-s + 0.883·32-s − 0.934·33-s − 0.217·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$8281$$    =    $$7^{2} \cdot 13^{2}$$ Sign: $-1$ Analytic conductor: $$66.1241$$ Root analytic conductor: $$8.13167$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{8281} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 8281,\ (\ :1/2),\ -1)$$

## Particular Values

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

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1$$
13 $$1$$
good2 $$1 - T + 2T^{2}$$
3 $$1 - 1.41T + 3T^{2}$$
5 $$1 - 4.09T + 5T^{2}$$
11 $$1 + 3.79T + 11T^{2}$$
17 $$1 + 1.26T + 17T^{2}$$
19 $$1 + 2.82T + 19T^{2}$$
23 $$1 + 7.79T + 23T^{2}$$
29 $$1 - 0.795T + 29T^{2}$$
31 $$1 + 1.41T + 31T^{2}$$
37 $$1 - 2.79T + 37T^{2}$$
41 $$1 - 2.97T + 41T^{2}$$
43 $$1 - 7.79T + 43T^{2}$$
47 $$1 - 2.82T + 47T^{2}$$
53 $$1 + 12.5T + 53T^{2}$$
59 $$1 + 12.4T + 59T^{2}$$
61 $$1 - 8.34T + 61T^{2}$$
67 $$1 + 3.79T + 67T^{2}$$
71 $$1 - 6T + 71T^{2}$$
73 $$1 + 12.5T + 73T^{2}$$
79 $$1 + 2.20T + 79T^{2}$$
83 $$1 + 9.89T + 83T^{2}$$
89 $$1 + 14.9T + 89T^{2}$$
97 $$1 + 4.24T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$