# Properties

 Label 4-112e3-1.1-c1e2-0-22 Degree $4$ Conductor $1404928$ Sign $-1$ Analytic cond. $89.5794$ Root an. cond. $3.07646$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 + 7-s − 2·9-s − 4·11-s + 4·23-s − 2·25-s + 2·29-s − 6·37-s − 4·43-s + 49-s + 2·53-s − 2·63-s + 12·67-s + 12·71-s − 4·77-s − 4·79-s − 5·81-s + 8·99-s + 20·107-s + 18·109-s + 4·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
 L(s)  = 1 + 0.377·7-s − 2/3·9-s − 1.20·11-s + 0.834·23-s − 2/5·25-s + 0.371·29-s − 0.986·37-s − 0.609·43-s + 1/7·49-s + 0.274·53-s − 0.251·63-s + 1.46·67-s + 1.42·71-s − 0.455·77-s − 0.450·79-s − 5/9·81-s + 0.804·99-s + 1.93·107-s + 1.72·109-s + 0.376·113-s − 0.909·121-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 + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1404928$$    =    $$2^{12} \cdot 7^{3}$$ Sign: $-1$ Analytic conductor: $$89.5794$$ Root analytic conductor: $$3.07646$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 1404928,\ (\ :1/2, 1/2),\ -1)$$

## 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 $$1$$
7$C_1$ $$1 - T$$
good3$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
5$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
11$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
13$C_2^2$ $$1 + 6 T^{2} + p^{2} T^{4}$$
17$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
19$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
23$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + p T^{2} )$$
29$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + p T^{2} )$$
31$C_2^2$ $$1 + 34 T^{2} + p^{2} T^{4}$$
37$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
41$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
47$C_2^2$ $$1 + 34 T^{2} + p^{2} T^{4}$$
53$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
59$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
61$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
67$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + p T^{2} )$$
71$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} )$$
73$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
79$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
83$C_2^2$ $$1 + 122 T^{2} + p^{2} T^{4}$$
89$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
97$C_2^2$ $$1 + 22 T^{2} + p^{2} T^{4}$$
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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

−7.902849194529290128412066826696, −7.29771736574378082140547406980, −6.93338841423648597738653698535, −6.40387682692778225404343628911, −5.97540858064662661225859173280, −5.33337199255747090866852779244, −5.16645654216811853848325438837, −4.78779887440354075389435368635, −4.05602568304114680384642713838, −3.52619923944422278802912308611, −3.00436983683295524022290162199, −2.45898666253051582326439656268, −1.95739283237888576654617304384, −1.01702583958425090641474605837, 0, 1.01702583958425090641474605837, 1.95739283237888576654617304384, 2.45898666253051582326439656268, 3.00436983683295524022290162199, 3.52619923944422278802912308611, 4.05602568304114680384642713838, 4.78779887440354075389435368635, 5.16645654216811853848325438837, 5.33337199255747090866852779244, 5.97540858064662661225859173280, 6.40387682692778225404343628911, 6.93338841423648597738653698535, 7.29771736574378082140547406980, 7.902849194529290128412066826696