# Properties

 Label 4-1008e2-1.1-c5e2-0-13 Degree $4$ Conductor $1016064$ Sign $1$ Analytic cond. $26136.1$ Root an. cond. $12.7148$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 42·5-s + 98·7-s − 716·11-s − 714·13-s + 1.34e3·17-s + 1.94e3·19-s − 1.79e3·23-s − 102·25-s + 1.20e3·29-s + 6.80e3·31-s − 4.11e3·35-s + 1.46e4·37-s − 7.89e3·41-s − 524·43-s − 1.83e4·47-s + 7.20e3·49-s − 4.51e4·53-s + 3.00e4·55-s + 2.25e4·59-s − 5.28e4·61-s + 2.99e4·65-s − 9.84e3·67-s − 840·71-s − 1.22e5·73-s − 7.01e4·77-s − 3.17e4·79-s + 3.69e4·83-s + ⋯
 L(s)  = 1 − 0.751·5-s + 0.755·7-s − 1.78·11-s − 1.17·13-s + 1.12·17-s + 1.23·19-s − 0.706·23-s − 0.0326·25-s + 0.264·29-s + 1.27·31-s − 0.567·35-s + 1.75·37-s − 0.733·41-s − 0.0432·43-s − 1.21·47-s + 3/7·49-s − 2.20·53-s + 1.34·55-s + 0.844·59-s − 1.81·61-s + 0.880·65-s − 0.268·67-s − 0.0197·71-s − 2.68·73-s − 1.34·77-s − 0.571·79-s + 0.589·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1016064$$    =    $$2^{8} \cdot 3^{4} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$26136.1$$ Root analytic conductor: $$12.7148$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 1016064,\ (\ :5/2, 5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ 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 $$1$$
7$C_1$ $$( 1 - p^{2} T )^{2}$$
good5$D_{4}$ $$1 + 42 T + 1866 T^{2} + 42 p^{5} T^{3} + p^{10} T^{4}$$
11$D_{4}$ $$1 + 716 T + 412438 T^{2} + 716 p^{5} T^{3} + p^{10} T^{4}$$
13$D_{4}$ $$1 + 714 T + 814258 T^{2} + 714 p^{5} T^{3} + p^{10} T^{4}$$
17$D_{4}$ $$1 - 1344 T + 2728510 T^{2} - 1344 p^{5} T^{3} + p^{10} T^{4}$$
19$D_{4}$ $$1 - 1946 T + 4229670 T^{2} - 1946 p^{5} T^{3} + p^{10} T^{4}$$
23$D_{4}$ $$1 + 1792 T + 11254510 T^{2} + 1792 p^{5} T^{3} + p^{10} T^{4}$$
29$D_{4}$ $$1 - 1200 T - 16154090 T^{2} - 1200 p^{5} T^{3} + p^{10} T^{4}$$
31$D_{4}$ $$1 - 6804 T + 30441118 T^{2} - 6804 p^{5} T^{3} + p^{10} T^{4}$$
37$D_{4}$ $$1 - 14640 T + 187693126 T^{2} - 14640 p^{5} T^{3} + p^{10} T^{4}$$
41$D_{4}$ $$1 + 7896 T + 209593854 T^{2} + 7896 p^{5} T^{3} + p^{10} T^{4}$$
43$D_{4}$ $$1 + 524 T + 242298998 T^{2} + 524 p^{5} T^{3} + p^{10} T^{4}$$
47$D_{4}$ $$1 + 18396 T + 435885630 T^{2} + 18396 p^{5} T^{3} + p^{10} T^{4}$$
53$D_{4}$ $$1 + 45132 T + 1149514990 T^{2} + 45132 p^{5} T^{3} + p^{10} T^{4}$$
59$D_{4}$ $$1 - 22582 T + 1531622854 T^{2} - 22582 p^{5} T^{3} + p^{10} T^{4}$$
61$D_{4}$ $$1 + 52822 T + 2145929546 T^{2} + 52822 p^{5} T^{3} + p^{10} T^{4}$$
67$D_{4}$ $$1 + 9848 T + 2714812022 T^{2} + 9848 p^{5} T^{3} + p^{10} T^{4}$$
71$D_{4}$ $$1 + 840 T + 427300302 T^{2} + 840 p^{5} T^{3} + p^{10} T^{4}$$
73$D_{4}$ $$1 + 122052 T + 7590539974 T^{2} + 122052 p^{5} T^{3} + p^{10} T^{4}$$
79$D_{4}$ $$1 + 31704 T + 6042098590 T^{2} + 31704 p^{5} T^{3} + p^{10} T^{4}$$
83$D_{4}$ $$1 - 36974 T + 4839605030 T^{2} - 36974 p^{5} T^{3} + p^{10} T^{4}$$
89$D_{4}$ $$1 - 210588 T + 21813719542 T^{2} - 210588 p^{5} T^{3} + p^{10} T^{4}$$
97$D_{4}$ $$1 + 44240 T + 2438219582 T^{2} + 44240 p^{5} T^{3} + p^{10} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$