# Properties

 Label 4-384e2-1.1-c1e2-0-7 Degree $4$ Conductor $147456$ Sign $1$ Analytic cond. $9.40192$ Root an. cond. $1.75107$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 3·9-s + 4·13-s + 6·25-s + 4·37-s − 10·49-s + 20·61-s − 4·73-s + 9·81-s + 4·97-s + 36·109-s − 12·117-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
 L(s)  = 1 − 9-s + 1.10·13-s + 6/5·25-s + 0.657·37-s − 1.42·49-s + 2.56·61-s − 0.468·73-s + 81-s + 0.406·97-s + 3.44·109-s − 1.10·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$147456$$    =    $$2^{14} \cdot 3^{2}$$ Sign: $1$ Analytic conductor: $$9.40192$$ Root analytic conductor: $$1.75107$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 147456,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.523255234$$ $$L(\frac12)$$ $$\approx$$ $$1.523255234$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
3$C_2$ $$1 + p T^{2}$$
good5$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
7$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
11$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
13$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
17$C_2^2$ $$1 - 18 T^{2} + p^{2} T^{4}$$
19$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
23$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
29$C_2^2$ $$1 - 22 T^{2} + p^{2} T^{4}$$
31$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
37$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
41$C_2^2$ $$1 + 62 T^{2} + p^{2} T^{4}$$
43$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
47$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
53$C_2^2$ $$1 - 70 T^{2} + p^{2} T^{4}$$
59$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
61$C_2$ $$( 1 - 10 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
83$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
89$C_2^2$ $$1 - 114 T^{2} + p^{2} T^{4}$$
97$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

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

−9.233890671639896233698812974001, −8.707309993624624316071067872564, −8.401054929572839837965369060406, −8.039834928723823082897119581940, −7.31132034306163719684988388577, −6.81527825984527674470556942561, −6.22921185260993032609162294248, −5.91621389219631449374969979452, −5.23327630659243614437718863728, −4.77673495577676993621153551843, −4.00070046643443942205357110876, −3.37334832115086858945784144332, −2.85339597931009576480664859577, −1.98823852106750385632031267224, −0.862419388998346649810443800729, 0.862419388998346649810443800729, 1.98823852106750385632031267224, 2.85339597931009576480664859577, 3.37334832115086858945784144332, 4.00070046643443942205357110876, 4.77673495577676993621153551843, 5.23327630659243614437718863728, 5.91621389219631449374969979452, 6.22921185260993032609162294248, 6.81527825984527674470556942561, 7.31132034306163719684988388577, 8.039834928723823082897119581940, 8.401054929572839837965369060406, 8.707309993624624316071067872564, 9.233890671639896233698812974001