# Properties

 Label 4-22e2-1.1-c9e2-0-0 Degree $4$ Conductor $484$ Sign $1$ Analytic cond. $128.386$ Root an. cond. $3.36612$ Motivic weight $9$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 32·2-s − 21·3-s + 768·4-s − 521·5-s + 672·6-s − 7.49e3·7-s − 1.63e4·8-s − 2.10e4·9-s + 1.66e4·10-s + 2.92e4·11-s − 1.61e4·12-s + 1.50e5·13-s + 2.39e5·14-s + 1.09e4·15-s + 3.27e5·16-s + 6.90e5·17-s + 6.73e5·18-s + 5.11e5·19-s − 4.00e5·20-s + 1.57e5·21-s − 9.37e5·22-s + 8.74e5·23-s + 3.44e5·24-s − 1.61e6·25-s − 4.81e6·26-s + 4.79e5·27-s − 5.75e6·28-s + ⋯
 L(s)  = 1 − 1.41·2-s − 0.149·3-s + 3/2·4-s − 0.372·5-s + 0.211·6-s − 1.17·7-s − 1.41·8-s − 1.06·9-s + 0.527·10-s + 0.603·11-s − 0.224·12-s + 1.45·13-s + 1.66·14-s + 0.0558·15-s + 5/4·16-s + 2.00·17-s + 1.51·18-s + 0.899·19-s − 0.559·20-s + 0.176·21-s − 0.852·22-s + 0.651·23-s + 0.211·24-s − 0.825·25-s − 2.06·26-s + 0.173·27-s − 1.76·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$484$$    =    $$2^{2} \cdot 11^{2}$$ Sign: $1$ Analytic conductor: $$128.386$$ Root analytic conductor: $$3.36612$$ Motivic weight: $$9$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 484,\ (\ :9/2, 9/2),\ 1)$$

## Particular Values

 $$L(5)$$ $$\approx$$ $$0.7042232530$$ $$L(\frac12)$$ $$\approx$$ $$0.7042232530$$ $$L(\frac{11}{2})$$ 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$C_1$ $$( 1 + p^{4} T )^{2}$$
11$C_1$ $$( 1 - p^{4} T )^{2}$$
good3$D_{4}$ $$1 + 7 p T + 2386 p^{2} T^{2} + 7 p^{10} T^{3} + p^{18} T^{4}$$
5$D_{4}$ $$1 + 521 T + 376592 p T^{2} + 521 p^{9} T^{3} + p^{18} T^{4}$$
7$D_{4}$ $$1 + 1070 p T + 10075602 p T^{2} + 1070 p^{10} T^{3} + p^{18} T^{4}$$
13$D_{4}$ $$1 - 150314 T + 26804211170 T^{2} - 150314 p^{9} T^{3} + p^{18} T^{4}$$
17$D_{4}$ $$1 - 40616 p T + 347226119086 T^{2} - 40616 p^{10} T^{3} + p^{18} T^{4}$$
19$D_{4}$ $$1 - 511212 T + 409533659590 T^{2} - 511212 p^{9} T^{3} + p^{18} T^{4}$$
23$D_{4}$ $$1 - 874751 T + 3350341409974 T^{2} - 874751 p^{9} T^{3} + p^{18} T^{4}$$
29$D_{4}$ $$1 + 2951058 T + 17050144887898 T^{2} + 2951058 p^{9} T^{3} + p^{18} T^{4}$$
31$D_{4}$ $$1 + 5818705 T + 31374188947838 T^{2} + 5818705 p^{9} T^{3} + p^{18} T^{4}$$
37$D_{4}$ $$1 - 2658905 T + 225492089266940 T^{2} - 2658905 p^{9} T^{3} + p^{18} T^{4}$$
41$D_{4}$ $$1 - 13427994 T + 688202282720050 T^{2} - 13427994 p^{9} T^{3} + p^{18} T^{4}$$
43$D_{4}$ $$1 + 17820762 T + 804875927328526 T^{2} + 17820762 p^{9} T^{3} + p^{18} T^{4}$$
47$D_{4}$ $$1 - 56044104 T + 2070094956889822 T^{2} - 56044104 p^{9} T^{3} + p^{18} T^{4}$$
53$D_{4}$ $$1 - 96842752 T + 7465684657276486 T^{2} - 96842752 p^{9} T^{3} + p^{18} T^{4}$$
59$D_{4}$ $$1 + 119136183 T + 5810749115433730 T^{2} + 119136183 p^{9} T^{3} + p^{18} T^{4}$$
61$D_{4}$ $$1 + 1482366 p T - 2562416206425278 T^{2} + 1482366 p^{10} T^{3} + p^{18} T^{4}$$
67$D_{4}$ $$1 + 295944891 T + 69921832215701962 T^{2} + 295944891 p^{9} T^{3} + p^{18} T^{4}$$
71$D_{4}$ $$1 + 322953267 T + 117737224590230422 T^{2} + 322953267 p^{9} T^{3} + p^{18} T^{4}$$
73$D_{4}$ $$1 + 255975514 T + 133557087434677034 T^{2} + 255975514 p^{9} T^{3} + p^{18} T^{4}$$
79$D_{4}$ $$1 + 889658 T + 151981069000373118 T^{2} + 889658 p^{9} T^{3} + p^{18} T^{4}$$
83$D_{4}$ $$1 + 277699042 T + 33227646296175022 T^{2} + 277699042 p^{9} T^{3} + p^{18} T^{4}$$
89$D_{4}$ $$1 - 1363672217 T + 1117996341977241880 T^{2} - 1363672217 p^{9} T^{3} + p^{18} T^{4}$$
97$D_{4}$ $$1 - 1398434043 T + 1658486726134145236 T^{2} - 1398434043 p^{9} T^{3} + p^{18} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$