# Properties

 Label 2-8280-1.1-c1-0-109 Degree $2$ Conductor $8280$ Sign $-1$ Analytic cond. $66.1161$ Root an. cond. $8.13118$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5-s + 2.34·7-s − 0.333·11-s + 3.37·13-s − 7.81·17-s + 1.20·19-s + 23-s + 25-s − 1.35·29-s − 10.2·31-s + 2.34·35-s + 2.83·37-s − 7.94·41-s − 7.92·43-s − 11.7·47-s − 1.47·49-s − 7.92·53-s − 0.333·55-s − 13.2·59-s + 2.88·61-s + 3.37·65-s + 7.70·67-s + 15.1·71-s − 7.02·73-s − 0.784·77-s + 0.325·79-s − 10.4·83-s + ⋯
 L(s)  = 1 + 0.447·5-s + 0.888·7-s − 0.100·11-s + 0.937·13-s − 1.89·17-s + 0.277·19-s + 0.208·23-s + 0.200·25-s − 0.251·29-s − 1.84·31-s + 0.397·35-s + 0.466·37-s − 1.24·41-s − 1.20·43-s − 1.71·47-s − 0.211·49-s − 1.08·53-s − 0.0450·55-s − 1.72·59-s + 0.369·61-s + 0.419·65-s + 0.941·67-s + 1.80·71-s − 0.821·73-s − 0.0893·77-s + 0.0365·79-s − 1.15·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$8280$$    =    $$2^{3} \cdot 3^{2} \cdot 5 \cdot 23$$ Sign: $-1$ Analytic conductor: $$66.1161$$ Root analytic conductor: $$8.13118$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{8280} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 8280,\ (\ :1/2),\ -1)$$

## 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$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1 - T$$
23 $$1 - T$$
good7 $$1 - 2.34T + 7T^{2}$$
11 $$1 + 0.333T + 11T^{2}$$
13 $$1 - 3.37T + 13T^{2}$$
17 $$1 + 7.81T + 17T^{2}$$
19 $$1 - 1.20T + 19T^{2}$$
29 $$1 + 1.35T + 29T^{2}$$
31 $$1 + 10.2T + 31T^{2}$$
37 $$1 - 2.83T + 37T^{2}$$
41 $$1 + 7.94T + 41T^{2}$$
43 $$1 + 7.92T + 43T^{2}$$
47 $$1 + 11.7T + 47T^{2}$$
53 $$1 + 7.92T + 53T^{2}$$
59 $$1 + 13.2T + 59T^{2}$$
61 $$1 - 2.88T + 61T^{2}$$
67 $$1 - 7.70T + 67T^{2}$$
71 $$1 - 15.1T + 71T^{2}$$
73 $$1 + 7.02T + 73T^{2}$$
79 $$1 - 0.325T + 79T^{2}$$
83 $$1 + 10.4T + 83T^{2}$$
89 $$1 + 8.05T + 89T^{2}$$
97 $$1 + 1.00T + 97T^{2}$$
show less
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−7.47666355126852599396980498866, −6.65103439372203577040503644488, −6.22029735915749709846471033980, −5.19764256268268685869102449995, −4.82996407597203751085274565178, −3.90330060302934983104759306787, −3.11006915028375604249485729118, −1.95025456946046942060788146022, −1.55090063688615433536504416990, 0, 1.55090063688615433536504416990, 1.95025456946046942060788146022, 3.11006915028375604249485729118, 3.90330060302934983104759306787, 4.82996407597203751085274565178, 5.19764256268268685869102449995, 6.22029735915749709846471033980, 6.65103439372203577040503644488, 7.47666355126852599396980498866