# Properties

 Label 4-28e4-1.1-c5e2-0-4 Degree $4$ Conductor $614656$ Sign $1$ Analytic cond. $15810.7$ Root an. cond. $11.2134$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 8·3-s − 38·5-s − 401·9-s − 424·11-s − 924·13-s − 304·15-s − 2.34e3·17-s + 360·19-s + 12·23-s − 1.46e3·25-s − 4.98e3·27-s − 7.05e3·29-s − 3.54e3·31-s − 3.39e3·33-s + 1.10e4·37-s − 7.39e3·39-s + 3.50e3·41-s + 1.26e4·43-s + 1.52e4·45-s + 2.29e4·47-s − 1.87e4·51-s + 3.04e3·53-s + 1.61e4·55-s + 2.88e3·57-s + 6.58e4·59-s − 4.24e4·61-s + 3.51e4·65-s + ⋯
 L(s)  = 1 + 0.513·3-s − 0.679·5-s − 1.65·9-s − 1.05·11-s − 1.51·13-s − 0.348·15-s − 1.96·17-s + 0.228·19-s + 0.00473·23-s − 0.469·25-s − 1.31·27-s − 1.55·29-s − 0.663·31-s − 0.542·33-s + 1.33·37-s − 0.778·39-s + 0.325·41-s + 1.04·43-s + 1.12·45-s + 1.51·47-s − 1.01·51-s + 0.148·53-s + 0.718·55-s + 0.117·57-s + 2.46·59-s − 1.46·61-s + 1.03·65-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$614656$$    =    $$2^{8} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$15810.7$$ Root analytic conductor: $$11.2134$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 614656,\ (\ :5/2, 5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.9397006712$$ $$L(\frac12)$$ $$\approx$$ $$0.9397006712$$ $$L(\frac{7}{2})$$ 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 $$1$$
7 $$1$$
good3$D_{4}$ $$1 - 8 T + 155 p T^{2} - 8 p^{5} T^{3} + p^{10} T^{4}$$
5$D_{4}$ $$1 + 38 T + 2911 T^{2} + 38 p^{5} T^{3} + p^{10} T^{4}$$
11$D_{4}$ $$1 + 424 T + 347473 T^{2} + 424 p^{5} T^{3} + p^{10} T^{4}$$
13$D_{4}$ $$1 + 924 T + 927022 T^{2} + 924 p^{5} T^{3} + p^{10} T^{4}$$
17$D_{4}$ $$1 + 138 p T + 3570955 T^{2} + 138 p^{6} T^{3} + p^{10} T^{4}$$
19$D_{4}$ $$1 - 360 T + 2145625 T^{2} - 360 p^{5} T^{3} + p^{10} T^{4}$$
23$D_{4}$ $$1 - 12 T + 12696565 T^{2} - 12 p^{5} T^{3} + p^{10} T^{4}$$
29$D_{4}$ $$1 + 7052 T + 35324974 T^{2} + 7052 p^{5} T^{3} + p^{10} T^{4}$$
31$D_{4}$ $$1 + 3548 T + 41490053 T^{2} + 3548 p^{5} T^{3} + p^{10} T^{4}$$
37$D_{4}$ $$1 - 11090 T + 146343239 T^{2} - 11090 p^{5} T^{3} + p^{10} T^{4}$$
41$D_{4}$ $$1 - 3500 T + 206898214 T^{2} - 3500 p^{5} T^{3} + p^{10} T^{4}$$
43$D_{4}$ $$1 - 12680 T + 267378054 T^{2} - 12680 p^{5} T^{3} + p^{10} T^{4}$$
47$D_{4}$ $$1 - 22956 T + 491525173 T^{2} - 22956 p^{5} T^{3} + p^{10} T^{4}$$
53$D_{4}$ $$1 - 3042 T + 716414839 T^{2} - 3042 p^{5} T^{3} + p^{10} T^{4}$$
59$D_{4}$ $$1 - 65808 T + 2502089257 T^{2} - 65808 p^{5} T^{3} + p^{10} T^{4}$$
61$D_{4}$ $$1 + 42486 T + 1501996159 T^{2} + 42486 p^{5} T^{3} + p^{10} T^{4}$$
67$D_{4}$ $$1 + 42312 T + 3116577793 T^{2} + 42312 p^{5} T^{3} + p^{10} T^{4}$$
71$D_{4}$ $$1 - 2208 T + 3433192846 T^{2} - 2208 p^{5} T^{3} + p^{10} T^{4}$$
73$D_{4}$ $$1 + 50506 T + 2773040987 T^{2} + 50506 p^{5} T^{3} + p^{10} T^{4}$$
79$D_{4}$ $$1 + 9004 T + 5176592589 T^{2} + 9004 p^{5} T^{3} + p^{10} T^{4}$$
83$D_{4}$ $$1 - 104328 T + 9837878230 T^{2} - 104328 p^{5} T^{3} + p^{10} T^{4}$$
89$D_{4}$ $$1 + 26666 T + 8107102555 T^{2} + 26666 p^{5} T^{3} + p^{10} T^{4}$$
97$D_{4}$ $$1 - 2156 p T + 28107307478 T^{2} - 2156 p^{6} T^{3} + p^{10} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$