# Properties

 Label 4-21e2-1.1-c22e2-0-0 Degree $4$ Conductor $441$ Sign $1$ Analytic cond. $4148.46$ Root an. cond. $8.02549$ Motivic weight $22$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 1.77e5·3-s − 4.19e6·4-s + 1.99e9·7-s − 7.43e11·12-s + 4.18e12·13-s − 2.06e13·19-s + 3.53e14·21-s − 2.38e15·25-s − 5.55e15·27-s − 8.37e15·28-s − 3.70e16·31-s − 8.44e16·37-s + 7.42e17·39-s + 2.53e18·43-s + 8.19e16·49-s − 1.75e19·52-s − 3.66e18·57-s + 6.73e19·61-s + 7.37e19·64-s − 2.14e20·67-s − 6.21e20·73-s − 4.22e20·75-s + 8.67e19·76-s + 1.03e21·79-s − 9.84e20·81-s − 1.48e21·84-s + 8.36e21·91-s + ⋯
 L(s)  = 1 + 3-s − 4-s + 1.01·7-s − 12-s + 2.33·13-s − 0.177·19-s + 1.01·21-s − 25-s − 27-s − 1.01·28-s − 1.45·31-s − 0.474·37-s + 2.33·39-s + 2.73·43-s + 0.0209·49-s − 2.33·52-s − 0.177·57-s + 1.54·61-s + 64-s − 1.75·67-s − 1.98·73-s − 75-s + 0.177·76-s + 1.38·79-s − 81-s − 1.01·84-s + 2.36·91-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(23-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+11)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$4148.46$$ Root analytic conductor: $$8.02549$$ Motivic weight: $$22$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{21} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 441,\ (\ :11, 11),\ 1)$$

## Particular Values

 $$L(\frac{23}{2})$$ $$\approx$$ $$3.046763515$$ $$L(\frac12)$$ $$\approx$$ $$3.046763515$$ $$L(12)$$ 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$C_2$ $$1 - p^{11} T + p^{22} T^{2}$$
7$C_2$ $$1 - 1997931923 T + p^{22} T^{2}$$
good2$C_2$ $$( 1 - p^{11} T + p^{22} T^{2} )( 1 + p^{11} T + p^{22} T^{2} )$$
5$C_2$ $$( 1 - p^{11} T + p^{22} T^{2} )( 1 + p^{11} T + p^{22} T^{2} )$$
11$C_2$ $$( 1 - p^{11} T + p^{22} T^{2} )( 1 + p^{11} T + p^{22} T^{2} )$$
13$C_2$ $$( 1 - 2094315786047 T + p^{22} T^{2} )^{2}$$
17$C_2$ $$( 1 - p^{11} T + p^{22} T^{2} )( 1 + p^{11} T + p^{22} T^{2} )$$
19$C_2$ $$( 1 - 190634130977363 T + p^{22} T^{2} )( 1 + 211308066581014 T + p^{22} T^{2} )$$
23$C_2$ $$( 1 - p^{11} T + p^{22} T^{2} )( 1 + p^{11} T + p^{22} T^{2} )$$
29$C_1$$\times$$C_1$ $$( 1 - p^{11} T )^{2}( 1 + p^{11} T )^{2}$$
31$C_2$ $$( 1 - 11550498557326034 T + p^{22} T^{2} )( 1 + 48632122495141621 T + p^{22} T^{2} )$$
37$C_2$ $$( 1 - 257128060064352074 T + p^{22} T^{2} )( 1 + 341584946762516377 T + p^{22} T^{2} )$$
41$C_1$$\times$$C_1$ $$( 1 - p^{11} T )^{2}( 1 + p^{11} T )^{2}$$
43$C_2$ $$( 1 - 1268503014678232811 T + p^{22} T^{2} )^{2}$$
47$C_2$ $$( 1 - p^{11} T + p^{22} T^{2} )( 1 + p^{11} T + p^{22} T^{2} )$$
53$C_2$ $$( 1 - p^{11} T + p^{22} T^{2} )( 1 + p^{11} T + p^{22} T^{2} )$$
59$C_2$ $$( 1 - p^{11} T + p^{22} T^{2} )( 1 + p^{11} T + p^{22} T^{2} )$$
61$C_2$ $$( 1 - 81382296126928348247 T + p^{22} T^{2} )( 1 + 13987693501999930801 T + p^{22} T^{2} )$$
67$C_2$ $$( 1 + 5287275662394476662 T + p^{22} T^{2} )( 1 +$$$$20\!\cdots\!41$$$$T + p^{22} T^{2} )$$
71$C_1$$\times$$C_1$ $$( 1 - p^{11} T )^{2}( 1 + p^{11} T )^{2}$$
73$C_2$ $$( 1 +$$$$23\!\cdots\!54$$$$T + p^{22} T^{2} )( 1 +$$$$38\!\cdots\!53$$$$T + p^{22} T^{2} )$$
79$C_2$ $$( 1 -$$$$14\!\cdots\!11$$$$T + p^{22} T^{2} )( 1 +$$$$41\!\cdots\!42$$$$T + p^{22} T^{2} )$$
83$C_1$$\times$$C_1$ $$( 1 - p^{11} T )^{2}( 1 + p^{11} T )^{2}$$
89$C_2$ $$( 1 - p^{11} T + p^{22} T^{2} )( 1 + p^{11} T + p^{22} T^{2} )$$
97$C_2$ $$( 1 +$$$$16\!\cdots\!42$$$$T + p^{22} T^{2} )^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−13.43683224837033743629287262401, −13.37517800005923698767915862728, −12.44653999021696057947110473897, −11.36621168039621992412375261579, −11.12816489753663890751407393301, −10.32887734908971639787694117015, −9.172345541004213088969310451312, −9.065689293554382311384110203527, −8.411835390081561262960752756214, −7.971420913561324854088479446579, −7.22247068488377560410173538599, −5.99321389419499714149232001011, −5.62837798200384204957780300273, −4.63385109819513583727593550331, −3.82347155259651740531913589461, −3.73641341379256289259871733339, −2.62096946047486632472174030516, −1.81991798293470538725445347445, −1.27399615008367403180032279431, −0.42523232211675483034127245658, 0.42523232211675483034127245658, 1.27399615008367403180032279431, 1.81991798293470538725445347445, 2.62096946047486632472174030516, 3.73641341379256289259871733339, 3.82347155259651740531913589461, 4.63385109819513583727593550331, 5.62837798200384204957780300273, 5.99321389419499714149232001011, 7.22247068488377560410173538599, 7.971420913561324854088479446579, 8.411835390081561262960752756214, 9.065689293554382311384110203527, 9.172345541004213088969310451312, 10.32887734908971639787694117015, 11.12816489753663890751407393301, 11.36621168039621992412375261579, 12.44653999021696057947110473897, 13.37517800005923698767915862728, 13.43683224837033743629287262401