# Properties

 Label 2-6e3-1.1-c3-0-7 Degree $2$ Conductor $216$ Sign $-1$ Analytic cond. $12.7444$ Root an. cond. $3.56993$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Learn more about

## Dirichlet series

 L(s)  = 1 − 15.4·5-s + 23.8·7-s + 14.2·11-s − 13.8·13-s − 80.5·17-s − 144.·19-s − 141.·23-s + 112.·25-s − 251.·29-s − 16.6·31-s − 367.·35-s + 305.·37-s + 429.·41-s − 181.·43-s − 79.4·47-s + 225.·49-s − 663.·53-s − 219.·55-s + 220.·59-s − 473.·61-s + 213.·65-s − 647.·67-s + 14.4·71-s + 776.·73-s + 339.·77-s − 257.·79-s + 1.28e3·83-s + ⋯
 L(s)  = 1 − 1.37·5-s + 1.28·7-s + 0.390·11-s − 0.295·13-s − 1.14·17-s − 1.74·19-s − 1.27·23-s + 0.901·25-s − 1.60·29-s − 0.0965·31-s − 1.77·35-s + 1.35·37-s + 1.63·41-s − 0.644·43-s − 0.246·47-s + 0.655·49-s − 1.72·53-s − 0.538·55-s + 0.486·59-s − 0.993·61-s + 0.406·65-s − 1.18·67-s + 0.0242·71-s + 1.24·73-s + 0.502·77-s − 0.367·79-s + 1.69·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$216$$    =    $$2^{3} \cdot 3^{3}$$ Sign: $-1$ Analytic conductor: $$12.7444$$ Root analytic conductor: $$3.56993$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{216} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 216,\ (\ :3/2),\ -1)$$

## 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$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5 $$1 + 15.4T + 125T^{2}$$
7 $$1 - 23.8T + 343T^{2}$$
11 $$1 - 14.2T + 1.33e3T^{2}$$
13 $$1 + 13.8T + 2.19e3T^{2}$$
17 $$1 + 80.5T + 4.91e3T^{2}$$
19 $$1 + 144.T + 6.85e3T^{2}$$
23 $$1 + 141.T + 1.21e4T^{2}$$
29 $$1 + 251.T + 2.43e4T^{2}$$
31 $$1 + 16.6T + 2.97e4T^{2}$$
37 $$1 - 305.T + 5.06e4T^{2}$$
41 $$1 - 429.T + 6.89e4T^{2}$$
43 $$1 + 181.T + 7.95e4T^{2}$$
47 $$1 + 79.4T + 1.03e5T^{2}$$
53 $$1 + 663.T + 1.48e5T^{2}$$
59 $$1 - 220.T + 2.05e5T^{2}$$
61 $$1 + 473.T + 2.26e5T^{2}$$
67 $$1 + 647.T + 3.00e5T^{2}$$
71 $$1 - 14.4T + 3.57e5T^{2}$$
73 $$1 - 776.T + 3.89e5T^{2}$$
79 $$1 + 257.T + 4.93e5T^{2}$$
83 $$1 - 1.28e3T + 5.71e5T^{2}$$
89 $$1 + 156.T + 7.04e5T^{2}$$
97 $$1 - 1.16e3T + 9.12e5T^{2}$$
show more
show less
$$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

−11.28681890177975284825942107145, −10.86272433452038570441576687664, −9.217849848799679557240575756936, −8.163597681983709554458218544042, −7.63205285410961500652380795349, −6.27406080109797370552182757789, −4.60455133020779375551468754608, −4.00875344980959931260724918236, −2.02675066573573264484141549007, 0, 2.02675066573573264484141549007, 4.00875344980959931260724918236, 4.60455133020779375551468754608, 6.27406080109797370552182757789, 7.63205285410961500652380795349, 8.163597681983709554458218544042, 9.217849848799679557240575756936, 10.86272433452038570441576687664, 11.28681890177975284825942107145