# Properties

 Label 4-2e16-1.1-c1e2-0-11 Degree $4$ Conductor $65536$ Sign $-1$ Analytic cond. $4.17863$ Root an. cond. $1.42974$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $1$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·9-s − 12·17-s − 10·25-s + 12·41-s − 14·49-s − 4·73-s − 5·81-s − 36·89-s + 20·97-s + 36·113-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
 L(s)  = 1 − 2/3·9-s − 2.91·17-s − 2·25-s + 1.87·41-s − 2·49-s − 0.468·73-s − 5/9·81-s − 3.81·89-s + 2.03·97-s + 3.38·113-s + 1.27·121-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 + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-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 & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$65536$$    =    $$2^{16}$$ Sign: $-1$ Analytic conductor: $$4.17863$$ Root analytic conductor: $$1.42974$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{65536} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 65536,\ (\ :1/2, 1/2),\ -1)$$

## 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$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
good3$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
5$C_2$ $$( 1 + p T^{2} )^{2}$$
7$C_2$ $$( 1 + p T^{2} )^{2}$$
11$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
13$C_2$ $$( 1 + p T^{2} )^{2}$$
17$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
23$C_2$ $$( 1 + p T^{2} )^{2}$$
29$C_2$ $$( 1 + p T^{2} )^{2}$$
31$C_2$ $$( 1 + p T^{2} )^{2}$$
37$C_2$ $$( 1 + p T^{2} )^{2}$$
41$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
47$C_2$ $$( 1 + p T^{2} )^{2}$$
53$C_2$ $$( 1 + p T^{2} )^{2}$$
59$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
61$C_2$ $$( 1 + p T^{2} )^{2}$$
67$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 + p T^{2} )^{2}$$
83$C_2$ $$( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} )$$
89$C_2$ $$( 1 + 18 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 - 10 T + p T^{2} )^{2}$$
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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

−9.739977345448053506810012798737, −9.084650424863502138877874475320, −8.559961689771162774439116698021, −8.337928305679105143801605101388, −7.49638267192940041493128754308, −7.14838005963241329022744903293, −6.29051795613585298614624812000, −6.13287781737238367476060833635, −5.43412469058090981748265870176, −4.43148007247498493089277458199, −4.38680180444175422137402538929, −3.38746738060370983088123271545, −2.49619173311123854201331757855, −1.90996633779170653420430529631, 0, 1.90996633779170653420430529631, 2.49619173311123854201331757855, 3.38746738060370983088123271545, 4.38680180444175422137402538929, 4.43148007247498493089277458199, 5.43412469058090981748265870176, 6.13287781737238367476060833635, 6.29051795613585298614624812000, 7.14838005963241329022744903293, 7.49638267192940041493128754308, 8.337928305679105143801605101388, 8.559961689771162774439116698021, 9.084650424863502138877874475320, 9.739977345448053506810012798737