# Properties

 Label 4-22e2-1.1-c13e2-0-1 Degree $4$ Conductor $484$ Sign $1$ Analytic cond. $556.526$ Root an. cond. $4.85703$ Motivic weight $13$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 128·2-s − 662·3-s + 1.22e4·4-s − 4.85e4·5-s + 8.47e4·6-s + 4.04e5·7-s − 1.04e6·8-s − 1.25e6·9-s + 6.21e6·10-s − 3.54e6·11-s − 8.13e6·12-s + 2.84e7·13-s − 5.17e7·14-s + 3.21e7·15-s + 8.38e7·16-s + 1.07e8·17-s + 1.61e8·18-s − 7.35e7·19-s − 5.96e8·20-s − 2.67e8·21-s + 4.53e8·22-s − 9.87e8·23-s + 6.94e8·24-s + 4.09e8·25-s − 3.64e9·26-s + 9.02e8·27-s + 4.97e9·28-s + ⋯
 L(s)  = 1 − 1.41·2-s − 0.524·3-s + 3/2·4-s − 1.39·5-s + 0.741·6-s + 1.29·7-s − 1.41·8-s − 0.789·9-s + 1.96·10-s − 0.603·11-s − 0.786·12-s + 1.63·13-s − 1.83·14-s + 0.728·15-s + 5/4·16-s + 1.07·17-s + 1.11·18-s − 0.358·19-s − 2.08·20-s − 0.681·21-s + 0.852·22-s − 1.39·23-s + 0.741·24-s + 0.335·25-s − 2.31·26-s + 0.448·27-s + 1.94·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$484$$    =    $$2^{2} \cdot 11^{2}$$ Sign: $1$ Analytic conductor: $$556.526$$ Root analytic conductor: $$4.85703$$ Motivic weight: $$13$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 484,\ (\ :13/2, 13/2),\ 1)$$

## Particular Values

 $$L(7)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{15}{2})$$ 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$C_1$ $$( 1 + p^{6} T )^{2}$$
11$C_1$ $$( 1 + p^{6} T )^{2}$$
good3$D_{4}$ $$1 + 662 T + 565861 p T^{2} + 662 p^{13} T^{3} + p^{26} T^{4}$$
5$D_{4}$ $$1 + 48566 T + 389809703 p T^{2} + 48566 p^{13} T^{3} + p^{26} T^{4}$$
7$D_{4}$ $$1 - 404508 T + 16233930190 p T^{2} - 404508 p^{13} T^{3} + p^{26} T^{4}$$
13$D_{4}$ $$1 - 28445952 T + 720464142541066 T^{2} - 28445952 p^{13} T^{3} + p^{26} T^{4}$$
17$D_{4}$ $$1 - 107032052 T + 20369477483479894 T^{2} - 107032052 p^{13} T^{3} + p^{26} T^{4}$$
19$D_{4}$ $$1 + 73524568 T + 79701138832491110 T^{2} + 73524568 p^{13} T^{3} + p^{26} T^{4}$$
23$D_{4}$ $$1 + 987164862 T + 1209740889736512703 T^{2} + 987164862 p^{13} T^{3} + p^{26} T^{4}$$
29$D_{4}$ $$1 + 9516399376 T + 42288055667555494426 T^{2} + 9516399376 p^{13} T^{3} + p^{26} T^{4}$$
31$D_{4}$ $$1 + 6865940430 T + 59716769884901551807 T^{2} + 6865940430 p^{13} T^{3} + p^{26} T^{4}$$
37$D_{4}$ $$1 + 4145233266 T +$$$$48\!\cdots\!59$$$$T^{2} + 4145233266 p^{13} T^{3} + p^{26} T^{4}$$
41$D_{4}$ $$1 - 39451523656 T +$$$$18\!\cdots\!10$$$$T^{2} - 39451523656 p^{13} T^{3} + p^{26} T^{4}$$
43$D_{4}$ $$1 - 5532549228 T +$$$$33\!\cdots\!86$$$$T^{2} - 5532549228 p^{13} T^{3} + p^{26} T^{4}$$
47$D_{4}$ $$1 + 28411609744 T +$$$$11\!\cdots\!74$$$$T^{2} + 28411609744 p^{13} T^{3} + p^{26} T^{4}$$
53$D_{4}$ $$1 + 49178542972 T +$$$$52\!\cdots\!42$$$$T^{2} + 49178542972 p^{13} T^{3} + p^{26} T^{4}$$
59$D_{4}$ $$1 - 110326136718 T +$$$$20\!\cdots\!95$$$$T^{2} - 110326136718 p^{13} T^{3} + p^{26} T^{4}$$
61$D_{4}$ $$1 + 161816895480 T +$$$$19\!\cdots\!62$$$$T^{2} + 161816895480 p^{13} T^{3} + p^{26} T^{4}$$
67$D_{4}$ $$1 + 483661746106 T +$$$$11\!\cdots\!27$$$$T^{2} + 483661746106 p^{13} T^{3} + p^{26} T^{4}$$
71$D_{4}$ $$1 - 318713704774 T +$$$$29\!\cdots\!95$$$$T^{2} - 318713704774 p^{13} T^{3} + p^{26} T^{4}$$
73$D_{4}$ $$1 + 1073843539168 T +$$$$30\!\cdots\!46$$$$T^{2} + 1073843539168 p^{13} T^{3} + p^{26} T^{4}$$
79$D_{4}$ $$1 + 742108796220 T +$$$$93\!\cdots\!22$$$$T^{2} + 742108796220 p^{13} T^{3} + p^{26} T^{4}$$
83$D_{4}$ $$1 + 6905249437156 T +$$$$29\!\cdots\!86$$$$T^{2} + 6905249437156 p^{13} T^{3} + p^{26} T^{4}$$
89$D_{4}$ $$1 + 8233927751362 T +$$$$60\!\cdots\!63$$$$T^{2} + 8233927751362 p^{13} T^{3} + p^{26} T^{4}$$
97$D_{4}$ $$1 + 4680888780654 T +$$$$18\!\cdots\!83$$$$T^{2} + 4680888780654 p^{13} T^{3} + p^{26} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$