# Properties

 Label 4-21e2-1.1-c31e2-0-0 Degree $4$ Conductor $441$ Sign $1$ Analytic cond. $16343.5$ Root an. cond. $11.3067$ Motivic weight $31$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4.30e7·3-s − 2.14e9·4-s + 9.53e12·7-s + 1.23e15·9-s − 9.24e16·12-s − 1.85e20·19-s + 4.10e20·21-s + 4.65e21·25-s + 2.65e22·27-s − 2.04e22·28-s + 2.74e23·31-s − 2.65e24·36-s + 2.25e24·37-s − 3.55e25·43-s − 6.68e25·49-s − 8.00e27·57-s + 1.62e28·61-s + 1.17e28·63-s + 9.90e27·64-s − 3.02e28·67-s + 2.29e29·73-s + 2.00e29·75-s + 3.99e29·76-s − 1.50e29·79-s + 3.81e29·81-s − 8.81e29·84-s + 1.18e31·93-s + ⋯
 L(s)  = 1 + 1.73·3-s − 4-s + 0.759·7-s + 2·9-s − 1.73·12-s − 2.80·19-s + 1.31·21-s + 25-s + 1.73·27-s − 0.759·28-s + 2.10·31-s − 2·36-s + 1.11·37-s − 1.70·43-s − 0.423·49-s − 4.86·57-s + 3.44·61-s + 1.51·63-s + 64-s − 1.50·67-s + 3.01·73-s + 1.73·75-s + 2.80·76-s − 0.580·79-s + 81-s − 1.31·84-s + 3.64·93-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(32-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+31/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$16343.5$$ Root analytic conductor: $$11.3067$$ Motivic weight: $$31$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{21} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 441,\ (\ :31/2, 31/2),\ 1)$$

## Particular Values

 $$L(16)$$ $$\approx$$ $$4.767387633$$ $$L(\frac12)$$ $$\approx$$ $$4.767387633$$ $$L(\frac{33}{2})$$ 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$C_2$ $$1 - p^{16} T + p^{31} T^{2}$$
7$C_2$ $$1 - 9534530086957 T + p^{31} T^{2}$$
good2$C_2^2$ $$1 + p^{31} T^{2} + p^{62} T^{4}$$
5$C_2^2$ $$1 - p^{31} T^{2} + p^{62} T^{4}$$
11$C_2^2$ $$1 + p^{31} T^{2} + p^{62} T^{4}$$
13$C_2$ $$( 1 - 121177137759872551 T + p^{31} T^{2} )( 1 + 121177137759872551 T + p^{31} T^{2} )$$
17$C_2^2$ $$1 - p^{31} T^{2} + p^{62} T^{4}$$
19$C_2$ $$( 1 + 54230691858525415549 T + p^{31} T^{2} )( 1 +$$$$13\!\cdots\!92$$$$T + p^{31} T^{2} )$$
23$C_2^2$ $$1 + p^{31} T^{2} + p^{62} T^{4}$$
29$C_2$ $$( 1 - p^{31} T^{2} )^{2}$$
31$C_2$ $$( 1 -$$$$24\!\cdots\!04$$$$T + p^{31} T^{2} )( 1 -$$$$33\!\cdots\!11$$$$T + p^{31} T^{2} )$$
37$C_2$ $$( 1 -$$$$40\!\cdots\!70$$$$T + p^{31} T^{2} )( 1 +$$$$17\!\cdots\!27$$$$T + p^{31} T^{2} )$$
41$C_2$ $$( 1 + p^{31} T^{2} )^{2}$$
43$C_2$ $$( 1 +$$$$17\!\cdots\!65$$$$T + p^{31} T^{2} )^{2}$$
47$C_2^2$ $$1 - p^{31} T^{2} + p^{62} T^{4}$$
53$C_2^2$ $$1 + p^{31} T^{2} + p^{62} T^{4}$$
59$C_2^2$ $$1 - p^{31} T^{2} + p^{62} T^{4}$$
61$C_2$ $$( 1 -$$$$86\!\cdots\!99$$$$T + p^{31} T^{2} )( 1 -$$$$75\!\cdots\!13$$$$T + p^{31} T^{2} )$$
67$C_2$ $$( 1 -$$$$78\!\cdots\!85$$$$T + p^{31} T^{2} )( 1 +$$$$38\!\cdots\!12$$$$T + p^{31} T^{2} )$$
71$C_2$ $$( 1 - p^{31} T^{2} )^{2}$$
73$C_2$ $$( 1 -$$$$15\!\cdots\!19$$$$T + p^{31} T^{2} )( 1 -$$$$76\!\cdots\!70$$$$T + p^{31} T^{2} )$$
79$C_2$ $$( 1 -$$$$35\!\cdots\!04$$$$T + p^{31} T^{2} )( 1 +$$$$50\!\cdots\!67$$$$T + p^{31} T^{2} )$$
83$C_2$ $$( 1 + p^{31} T^{2} )^{2}$$
89$C_2^2$ $$1 - p^{31} T^{2} + p^{62} T^{4}$$
97$C_2$ $$( 1 -$$$$73\!\cdots\!98$$$$T + p^{31} T^{2} )( 1 +$$$$73\!\cdots\!98$$$$T + p^{31} T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$