# Properties

 Label 2-168e2-1.1-c1-0-166 Degree $2$ Conductor $28224$ Sign $-1$ Analytic cond. $225.369$ Root an. cond. $15.0123$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 + 3·5-s + 11-s + 4·13-s − 4·17-s + 8·23-s + 4·25-s − 7·29-s − 11·31-s − 4·37-s + 4·41-s − 2·43-s − 2·47-s − 11·53-s + 3·55-s − 7·59-s − 10·61-s + 12·65-s + 10·67-s + 6·71-s − 6·73-s − 11·79-s − 11·83-s − 12·85-s − 6·89-s + 7·97-s + 101-s + 103-s + ⋯
 L(s)  = 1 + 1.34·5-s + 0.301·11-s + 1.10·13-s − 0.970·17-s + 1.66·23-s + 4/5·25-s − 1.29·29-s − 1.97·31-s − 0.657·37-s + 0.624·41-s − 0.304·43-s − 0.291·47-s − 1.51·53-s + 0.404·55-s − 0.911·59-s − 1.28·61-s + 1.48·65-s + 1.22·67-s + 0.712·71-s − 0.702·73-s − 1.23·79-s − 1.20·83-s − 1.30·85-s − 0.635·89-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$28224$$    =    $$2^{6} \cdot 3^{2} \cdot 7^{2}$$ Sign: $-1$ Analytic conductor: $$225.369$$ Root analytic conductor: $$15.0123$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 28224,\ (\ :1/2),\ -1)$$

## 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$$
7 $$1$$
good5 $$1 - 3 T + p T^{2}$$
11 $$1 - T + p T^{2}$$
13 $$1 - 4 T + p T^{2}$$
17 $$1 + 4 T + p T^{2}$$
19 $$1 + p T^{2}$$
23 $$1 - 8 T + p T^{2}$$
29 $$1 + 7 T + p T^{2}$$
31 $$1 + 11 T + p T^{2}$$
37 $$1 + 4 T + p T^{2}$$
41 $$1 - 4 T + p T^{2}$$
43 $$1 + 2 T + p T^{2}$$
47 $$1 + 2 T + p T^{2}$$
53 $$1 + 11 T + p T^{2}$$
59 $$1 + 7 T + p T^{2}$$
61 $$1 + 10 T + p T^{2}$$
67 $$1 - 10 T + p T^{2}$$
71 $$1 - 6 T + p T^{2}$$
73 $$1 + 6 T + p T^{2}$$
79 $$1 + 11 T + p T^{2}$$
83 $$1 + 11 T + p T^{2}$$
89 $$1 + 6 T + p T^{2}$$
97 $$1 - 7 T + p T^{2}$$
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$$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

−15.52678187675779, −14.86390389511094, −14.31784517999833, −13.92022343072964, −13.23578225252949, −12.93191457369054, −12.62775072571019, −11.45339943005617, −11.13501777176296, −10.75444891247928, −10.05697826375737, −9.247948091145523, −9.134929055758498, −8.655961185175661, −7.679245767587642, −7.108733599551109, −6.445831362790403, −6.039541854019007, −5.382297624350448, −4.894824068828006, −3.973456298608537, −3.362929756163732, −2.575927940053931, −1.694409940374000, −1.391676018247986, 0, 1.391676018247986, 1.694409940374000, 2.575927940053931, 3.362929756163732, 3.973456298608537, 4.894824068828006, 5.382297624350448, 6.039541854019007, 6.445831362790403, 7.108733599551109, 7.679245767587642, 8.655961185175661, 9.134929055758498, 9.247948091145523, 10.05697826375737, 10.75444891247928, 11.13501777176296, 11.45339943005617, 12.62775072571019, 12.93191457369054, 13.23578225252949, 13.92022343072964, 14.31784517999833, 14.86390389511094, 15.52678187675779