Properties

 Label 4-384e2-1.1-c1e2-0-5 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 $0$

Origins

Dirichlet series

 L(s)  = 1 − 3·9-s + 6·25-s + 14·49-s + 12·73-s + 9·81-s + 36·97-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 18·225-s + ⋯
 L(s)  = 1 − 9-s + 6/5·25-s + 2·49-s + 1.40·73-s + 81-s + 3.65·97-s − 2·121-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 + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s − 6/5·225-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: $\chi_{147456} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 147456,\ (\ :1/2, 1/2),\ 1)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$1.403391428$$ $$L(\frac12)$$ $$\approx$$ $$1.403391428$$ $$L(\frac{3}{2})$$ 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$C_2$ $$( 1 - p T^{2} )^{2}$$
11$C_2$ $$( 1 + p T^{2} )^{2}$$
13$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
17$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
19$C_2$ $$( 1 + p T^{2} )^{2}$$
23$C_2$ $$( 1 + p T^{2} )^{2}$$
29$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
31$C_2$ $$( 1 - p T^{2} )^{2}$$
37$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
41$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
43$C_2$ $$( 1 + p T^{2} )^{2}$$
47$C_2$ $$( 1 + p T^{2} )^{2}$$
53$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
59$C_2$ $$( 1 + p T^{2} )^{2}$$
61$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
67$C_2$ $$( 1 + p T^{2} )^{2}$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 - p T^{2} )^{2}$$
83$C_2$ $$( 1 + p T^{2} )^{2}$$
89$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
97$C_2$ $$( 1 - 18 T + p T^{2} )^{2}$$
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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

−9.131445194439497711423343193703, −8.810531425734125505560177559066, −8.510267167328683949354739212465, −7.73016249244442554423951279873, −7.53112331592469102778504224020, −6.70014507076017366407014120928, −6.42450463718800424724660339570, −5.73492123379650911132030342118, −5.28639122118936362306125751131, −4.77087759206350184572425319619, −4.03627088358252137665269239478, −3.38480863103310061241007005681, −2.75434798548514139439620711597, −2.08534696107705366345513395003, −0.809435294418029445368878604560, 0.809435294418029445368878604560, 2.08534696107705366345513395003, 2.75434798548514139439620711597, 3.38480863103310061241007005681, 4.03627088358252137665269239478, 4.77087759206350184572425319619, 5.28639122118936362306125751131, 5.73492123379650911132030342118, 6.42450463718800424724660339570, 6.70014507076017366407014120928, 7.53112331592469102778504224020, 7.73016249244442554423951279873, 8.510267167328683949354739212465, 8.810531425734125505560177559066, 9.131445194439497711423343193703