# Properties

 Label 4-112e3-1.1-c1e2-0-39 Degree $4$ Conductor $1404928$ Sign $1$ Analytic cond. $89.5794$ Root an. cond. $3.07646$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 7-s − 6·9-s − 8·11-s − 6·25-s − 12·29-s + 4·37-s − 8·43-s + 49-s − 12·53-s − 6·63-s − 8·67-s + 16·71-s − 8·77-s − 32·79-s + 27·81-s + 48·99-s − 24·107-s + 20·109-s + 4·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
 L(s)  = 1 + 0.377·7-s − 2·9-s − 2.41·11-s − 6/5·25-s − 2.22·29-s + 0.657·37-s − 1.21·43-s + 1/7·49-s − 1.64·53-s − 0.755·63-s − 0.977·67-s + 1.89·71-s − 0.911·77-s − 3.60·79-s + 3·81-s + 4.82·99-s − 2.32·107-s + 1.91·109-s + 0.376·113-s + 2.36·121-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 + ⋯

## 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: no Self-dual: yes Analytic rank: $$2$$ 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$C_2$ $$( 1 + p T^{2} )^{2}$$
5$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
11$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
13$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
19$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
23$C_2$ $$( 1 + p T^{2} )^{2}$$
29$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
31$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
37$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
43$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
47$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
53$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
59$C_2$ $$( 1 + p T^{2} )^{2}$$
61$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
67$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
71$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
79$C_2$ $$( 1 + 16 T + p T^{2} )^{2}$$
83$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
89$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
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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

−7.79292083571821938290976650763, −7.26382021666693896035142333427, −6.59264336057149769206055975154, −5.91740132798549001211232285063, −5.69394971603405508863657314834, −5.47493819124631815597461836959, −4.92225264716551837532790929786, −4.53211800578457486655010876487, −3.60173413237973468362473708205, −3.32773603072291404011890923635, −2.58182243115007516792385368988, −2.40145156924384381758797773878, −1.62407385510972851799092712087, 0, 0, 1.62407385510972851799092712087, 2.40145156924384381758797773878, 2.58182243115007516792385368988, 3.32773603072291404011890923635, 3.60173413237973468362473708205, 4.53211800578457486655010876487, 4.92225264716551837532790929786, 5.47493819124631815597461836959, 5.69394971603405508863657314834, 5.91740132798549001211232285063, 6.59264336057149769206055975154, 7.26382021666693896035142333427, 7.79292083571821938290976650763