# Properties

 Label 4-21e2-1.1-c13e2-0-1 Degree $4$ Conductor $441$ Sign $1$ Analytic cond. $507.082$ Root an. cond. $4.74536$ Motivic weight $13$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 144·2-s + 1.45e3·3-s + 1.52e3·4-s − 5.25e4·5-s + 2.09e5·6-s + 2.35e5·7-s − 1.36e6·8-s + 1.59e6·9-s − 7.56e6·10-s − 8.59e6·11-s + 2.21e6·12-s − 3.79e7·13-s + 3.38e7·14-s − 7.66e7·15-s − 1.23e8·16-s + 1.66e7·17-s + 2.29e8·18-s − 2.11e8·19-s − 7.98e7·20-s + 3.43e8·21-s − 1.23e9·22-s − 1.94e8·23-s − 1.99e9·24-s + 9.18e8·25-s − 5.46e9·26-s + 1.54e9·27-s + 3.57e8·28-s + ⋯
 L(s)  = 1 + 1.59·2-s + 1.15·3-s + 0.185·4-s − 1.50·5-s + 1.83·6-s + 0.755·7-s − 1.84·8-s + 9-s − 2.39·10-s − 1.46·11-s + 0.214·12-s − 2.18·13-s + 1.20·14-s − 1.73·15-s − 1.84·16-s + 0.167·17-s + 1.59·18-s − 1.02·19-s − 0.279·20-s + 0.872·21-s − 2.32·22-s − 0.274·23-s − 2.13·24-s + 0.752·25-s − 3.47·26-s + 0.769·27-s + 0.140·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$507.082$$ Root analytic conductor: $$4.74536$$ Motivic weight: $$13$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{21} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 441,\ (\ :13/2, 13/2),\ 1)$$

## 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)$
bad3$C_1$ $$( 1 - p^{6} T )^{2}$$
7$C_1$ $$( 1 - p^{6} T )^{2}$$
good2$D_{4}$ $$1 - 9 p^{4} T + 1201 p^{4} T^{2} - 9 p^{17} T^{3} + p^{26} T^{4}$$
5$D_{4}$ $$1 + 10512 p T + 73763578 p^{2} T^{2} + 10512 p^{14} T^{3} + p^{26} T^{4}$$
11$D_{4}$ $$1 + 781236 p T + 551670639946 p^{2} T^{2} + 781236 p^{14} T^{3} + p^{26} T^{4}$$
13$D_{4}$ $$1 + 37982012 T + 854172190564734 T^{2} + 37982012 p^{13} T^{3} + p^{26} T^{4}$$
17$D_{4}$ $$1 - 16643088 T + 4110551987575522 T^{2} - 16643088 p^{13} T^{3} + p^{26} T^{4}$$
19$D_{4}$ $$1 + 211195304 T + 37306293658457814 T^{2} + 211195304 p^{13} T^{3} + p^{26} T^{4}$$
23$D_{4}$ $$1 + 194997852 T + 640913258073572530 T^{2} + 194997852 p^{13} T^{3} + p^{26} T^{4}$$
29$D_{4}$ $$1 + 3481642548 T + 6322342576139078206 T^{2} + 3481642548 p^{13} T^{3} + p^{26} T^{4}$$
31$D_{4}$ $$1 + 8393797520 T + 47374586993368068990 T^{2} + 8393797520 p^{13} T^{3} + p^{26} T^{4}$$
37$D_{4}$ $$1 + 22337946716 T +$$$$56\!\cdots\!86$$$$T^{2} + 22337946716 p^{13} T^{3} + p^{26} T^{4}$$
41$D_{4}$ $$1 + 12507403320 T +$$$$12\!\cdots\!42$$$$T^{2} + 12507403320 p^{13} T^{3} + p^{26} T^{4}$$
43$D_{4}$ $$1 - 26062746328 T +$$$$32\!\cdots\!54$$$$T^{2} - 26062746328 p^{13} T^{3} + p^{26} T^{4}$$
47$D_{4}$ $$1 - 29222658936 T +$$$$49\!\cdots\!30$$$$T^{2} - 29222658936 p^{13} T^{3} + p^{26} T^{4}$$
53$D_{4}$ $$1 - 523911868308 T +$$$$12\!\cdots\!90$$$$T^{2} - 523911868308 p^{13} T^{3} + p^{26} T^{4}$$
59$D_{4}$ $$1 + 94444581024 T +$$$$21\!\cdots\!14$$$$T^{2} + 94444581024 p^{13} T^{3} + p^{26} T^{4}$$
61$D_{4}$ $$1 - 461720554252 T +$$$$31\!\cdots\!26$$$$T^{2} - 461720554252 p^{13} T^{3} + p^{26} T^{4}$$
67$D_{4}$ $$1 + 1237879942064 T +$$$$14\!\cdots\!30$$$$T^{2} + 1237879942064 p^{13} T^{3} + p^{26} T^{4}$$
71$D_{4}$ $$1 - 1961126619804 T +$$$$32\!\cdots\!26$$$$T^{2} - 1961126619804 p^{13} T^{3} + p^{26} T^{4}$$
73$D_{4}$ $$1 + 2407791783284 T +$$$$47\!\cdots\!62$$$$T^{2} + 2407791783284 p^{13} T^{3} + p^{26} T^{4}$$
79$D_{4}$ $$1 + 2622494223848 T +$$$$11\!\cdots\!42$$$$T^{2} + 2622494223848 p^{13} T^{3} + p^{26} T^{4}$$
83$D_{4}$ $$1 - 4489561815912 T +$$$$20\!\cdots\!90$$$$T^{2} - 4489561815912 p^{13} T^{3} + p^{26} T^{4}$$
89$D_{4}$ $$1 - 520927840584 T +$$$$43\!\cdots\!50$$$$T^{2} - 520927840584 p^{13} T^{3} + p^{26} T^{4}$$
97$D_{4}$ $$1 + 13027956631604 T +$$$$17\!\cdots\!86$$$$T^{2} + 13027956631604 p^{13} T^{3} + p^{26} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$