# Properties

 Label 4-22e2-1.1-c11e2-0-0 Degree $4$ Conductor $484$ Sign $1$ Analytic cond. $285.730$ Root an. cond. $4.11139$ Motivic weight $11$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 64·2-s − 426·3-s + 3.07e3·4-s + 2.29e3·5-s − 2.72e4·6-s − 8.63e4·7-s + 1.31e5·8-s − 2.75e4·9-s + 1.46e5·10-s − 3.22e5·11-s − 1.30e6·12-s − 2.10e6·13-s − 5.52e6·14-s − 9.75e5·15-s + 5.24e6·16-s − 2.88e6·17-s − 1.76e6·18-s − 1.95e7·19-s + 7.03e6·20-s + 3.67e7·21-s − 2.06e7·22-s + 1.26e7·23-s − 5.58e7·24-s − 4.07e7·25-s − 1.34e8·26-s + 2.53e7·27-s − 2.65e8·28-s + ⋯
 L(s)  = 1 + 1.41·2-s − 1.01·3-s + 3/2·4-s + 0.327·5-s − 1.43·6-s − 1.94·7-s + 1.41·8-s − 0.155·9-s + 0.463·10-s − 0.603·11-s − 1.51·12-s − 1.56·13-s − 2.74·14-s − 0.331·15-s + 5/4·16-s − 0.492·17-s − 0.219·18-s − 1.81·19-s + 0.491·20-s + 1.96·21-s − 0.852·22-s + 0.410·23-s − 1.43·24-s − 0.834·25-s − 2.21·26-s + 0.339·27-s − 2.91·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$484$$    =    $$2^{2} \cdot 11^{2}$$ Sign: $1$ Analytic conductor: $$285.730$$ Root analytic conductor: $$4.11139$$ Motivic weight: $$11$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 484,\ (\ :11/2, 11/2),\ 1)$$

## Particular Values

 $$L(6)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{13}{2})$$ 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$C_1$ $$( 1 - p^{5} T )^{2}$$
11$C_1$ $$( 1 + p^{5} T )^{2}$$
good3$D_{4}$ $$1 + 142 p T + 7741 p^{3} T^{2} + 142 p^{12} T^{3} + p^{22} T^{4}$$
5$D_{4}$ $$1 - 458 p T + 1840291 p^{2} T^{2} - 458 p^{12} T^{3} + p^{22} T^{4}$$
7$D_{4}$ $$1 + 12332 p T + 5240877330 T^{2} + 12332 p^{12} T^{3} + p^{22} T^{4}$$
13$D_{4}$ $$1 + 2100184 T + 3031952118074 T^{2} + 2100184 p^{11} T^{3} + p^{22} T^{4}$$
17$D_{4}$ $$1 + 2882276 T + 62188541127526 T^{2} + 2882276 p^{11} T^{3} + p^{22} T^{4}$$
19$D_{4}$ $$1 + 19571712 T + 327883644494230 T^{2} + 19571712 p^{11} T^{3} + p^{22} T^{4}$$
23$D_{4}$ $$1 - 12680534 T + 1705711199562247 T^{2} - 12680534 p^{11} T^{3} + p^{22} T^{4}$$
29$D_{4}$ $$1 - 45662496 T + 21179339599363786 T^{2} - 45662496 p^{11} T^{3} + p^{22} T^{4}$$
31$D_{4}$ $$1 + 506504170 T + 114187742824378487 T^{2} + 506504170 p^{11} T^{3} + p^{22} T^{4}$$
37$D_{4}$ $$1 - 402672518 T + 284017789497870851 T^{2} - 402672518 p^{11} T^{3} + p^{22} T^{4}$$
41$D_{4}$ $$1 - 608864016 T + 427507291180888690 T^{2} - 608864016 p^{11} T^{3} + p^{22} T^{4}$$
43$D_{4}$ $$1 - 1100094564 T + 776360461440501034 T^{2} - 1100094564 p^{11} T^{3} + p^{22} T^{4}$$
47$D_{4}$ $$1 - 1012342272 T + 709446719476370206 T^{2} - 1012342272 p^{11} T^{3} + p^{22} T^{4}$$
53$D_{4}$ $$1 + 68189276 T + 5575175257378167838 T^{2} + 68189276 p^{11} T^{3} + p^{22} T^{4}$$
59$D_{4}$ $$1 + 6791617518 T + 61459236505844701015 T^{2} + 6791617518 p^{11} T^{3} + p^{22} T^{4}$$
61$D_{4}$ $$1 - 5704046520 T + 26335031748690586522 T^{2} - 5704046520 p^{11} T^{3} + p^{22} T^{4}$$
67$D_{4}$ $$1 + 36514311702 T +$$$$57\!\cdots\!63$$$$T^{2} + 36514311702 p^{11} T^{3} + p^{22} T^{4}$$
71$D_{4}$ $$1 - 20672196594 T +$$$$28\!\cdots\!35$$$$T^{2} - 20672196594 p^{11} T^{3} + p^{22} T^{4}$$
73$D_{4}$ $$1 - 3082870856 T +$$$$57\!\cdots\!74$$$$T^{2} - 3082870856 p^{11} T^{3} + p^{22} T^{4}$$
79$D_{4}$ $$1 + 28681382180 T +$$$$17\!\cdots\!22$$$$T^{2} + 28681382180 p^{11} T^{3} + p^{22} T^{4}$$
83$D_{4}$ $$1 - 29532640772 T +$$$$26\!\cdots\!34$$$$T^{2} - 29532640772 p^{11} T^{3} + p^{22} T^{4}$$
89$D_{4}$ $$1 - 85063742462 T +$$$$73\!\cdots\!63$$$$T^{2} - 85063742462 p^{11} T^{3} + p^{22} T^{4}$$
97$D_{4}$ $$1 + 180832449678 T +$$$$22\!\cdots\!27$$$$T^{2} + 180832449678 p^{11} T^{3} + p^{22} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$