# Properties

 Label 4-2736e2-1.1-c1e2-0-38 Degree $4$ Conductor $7485696$ Sign $1$ Analytic cond. $477.294$ Root an. cond. $4.67408$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·5-s + 6·7-s + 8·11-s − 5·13-s + 8·19-s − 4·23-s + 5·25-s + 8·29-s − 2·31-s + 24·35-s − 10·37-s + 8·41-s − 5·43-s + 8·47-s + 13·49-s + 4·53-s + 32·55-s − 12·59-s + 61-s − 20·65-s + 3·67-s − 16·71-s + 15·73-s + 48·77-s − 7·79-s − 12·89-s − 30·91-s + ⋯
 L(s)  = 1 + 1.78·5-s + 2.26·7-s + 2.41·11-s − 1.38·13-s + 1.83·19-s − 0.834·23-s + 25-s + 1.48·29-s − 0.359·31-s + 4.05·35-s − 1.64·37-s + 1.24·41-s − 0.762·43-s + 1.16·47-s + 13/7·49-s + 0.549·53-s + 4.31·55-s − 1.56·59-s + 0.128·61-s − 2.48·65-s + 0.366·67-s − 1.89·71-s + 1.75·73-s + 5.47·77-s − 0.787·79-s − 1.27·89-s − 3.14·91-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$7485696$$    =    $$2^{8} \cdot 3^{4} \cdot 19^{2}$$ Sign: $1$ Analytic conductor: $$477.294$$ Root analytic conductor: $$4.67408$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 7485696,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$7.095042780$$ $$L(\frac12)$$ $$\approx$$ $$7.095042780$$ $$L(\frac{3}{2})$$ 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 $$1$$
19$C_2$ $$1 - 8 T + p T^{2}$$
good5$C_2^2$ $$1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
7$C_2$ $$( 1 - 3 T + p T^{2} )^{2}$$
11$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
13$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
17$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
23$C_2^2$ $$1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
29$C_2^2$ $$1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
31$C_2$ $$( 1 + T + p T^{2} )^{2}$$
37$C_2$ $$( 1 + 5 T + p T^{2} )^{2}$$
41$C_2^2$ $$1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
43$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} )$$
47$C_2^2$ $$1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
53$C_2^2$ $$1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
59$C_2^2$ $$1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
61$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} )$$
67$C_2^2$ $$1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
71$C_2^2$ $$1 + 16 T + 185 T^{2} + 16 p T^{3} + p^{2} T^{4}$$
73$C_2^2$ $$1 - 15 T + 152 T^{2} - 15 p T^{3} + p^{2} T^{4}$$
79$C_2^2$ $$1 + 7 T - 30 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
83$C_2$ $$( 1 + p T^{2} )^{2}$$
89$C_2^2$ $$1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
97$C_2^2$ $$1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
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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

−9.241431494376915632436297351459, −8.843196246827819270476044869076, −8.131730964228801101711620765473, −8.029850932224612115333896879655, −7.32817658501532653875286554678, −7.26598114923559937606200382408, −6.73556085055988780283667544223, −6.13993357742049147799532149302, −6.08866185893380106237319383682, −5.34167253995475635532829145136, −5.19549848884685461009015931338, −4.90668742092623487607894746040, −4.20094931029252274647204638510, −4.13608802513676338527472232413, −3.31222740790048070237403236388, −2.72696892334717888625116720639, −2.15422673641753045623852449635, −1.60887496822223174216876199303, −1.54929220257768350982780903308, −0.932705803029573658256122849041, 0.932705803029573658256122849041, 1.54929220257768350982780903308, 1.60887496822223174216876199303, 2.15422673641753045623852449635, 2.72696892334717888625116720639, 3.31222740790048070237403236388, 4.13608802513676338527472232413, 4.20094931029252274647204638510, 4.90668742092623487607894746040, 5.19549848884685461009015931338, 5.34167253995475635532829145136, 6.08866185893380106237319383682, 6.13993357742049147799532149302, 6.73556085055988780283667544223, 7.26598114923559937606200382408, 7.32817658501532653875286554678, 8.029850932224612115333896879655, 8.131730964228801101711620765473, 8.843196246827819270476044869076, 9.241431494376915632436297351459