# Properties

 Label 4-21e2-1.1-c15e2-0-0 Degree $4$ Conductor $441$ Sign $1$ Analytic cond. $897.939$ Root an. cond. $5.47408$ Motivic weight $15$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 6.55e4·4-s − 1.24e6·7-s − 1.43e7·9-s + 3.22e9·16-s − 6.10e10·25-s − 8.15e10·28-s − 9.40e11·36-s − 2.18e12·37-s − 2.88e12·43-s − 3.19e12·49-s + 1.78e13·63-s + 1.40e14·64-s + 1.98e14·67-s − 1.77e14·79-s + 2.05e14·81-s − 4.00e15·100-s + 7.56e15·109-s − 4.01e15·112-s + 8.35e15·121-s + 127-s + 131-s + 137-s + 139-s − 4.62e16·144-s − 1.42e17·148-s + 149-s + 151-s + ⋯
 L(s)  = 1 + 2·4-s − 0.571·7-s − 9-s + 3·16-s − 2·25-s − 1.14·28-s − 2·36-s − 3.77·37-s − 1.61·43-s − 0.673·49-s + 0.571·63-s + 4·64-s + 3.99·67-s − 1.03·79-s + 81-s − 4·100-s + 3.96·109-s − 1.71·112-s + 2·121-s − 3·144-s − 7.55·148-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+15/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$897.939$$ Root analytic conductor: $$5.47408$$ Motivic weight: $$15$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{21} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 441,\ (\ :15/2, 15/2),\ 1)$$

## Particular Values

 $$L(8)$$ $$\approx$$ $$2.749166253$$ $$L(\frac12)$$ $$\approx$$ $$2.749166253$$ $$L(\frac{17}{2})$$ 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$C_2$ $$1 + p^{15} T^{2}$$
7$C_2$ $$1 + 1244900 T + p^{15} T^{2}$$
good2$C_2$ $$( 1 - p^{15} T^{2} )^{2}$$
5$C_2$ $$( 1 + p^{15} T^{2} )^{2}$$
11$C_2$ $$( 1 - p^{15} T^{2} )^{2}$$
13$C_2$ $$( 1 - 397771850 T + p^{15} T^{2} )( 1 + 397771850 T + p^{15} T^{2} )$$
17$C_2$ $$( 1 + p^{15} T^{2} )^{2}$$
19$C_2$ $$( 1 - 7700827736 T + p^{15} T^{2} )( 1 + 7700827736 T + p^{15} T^{2} )$$
23$C_2$ $$( 1 - p^{15} T^{2} )^{2}$$
29$C_2$ $$( 1 - p^{15} T^{2} )^{2}$$
31$C_2$ $$( 1 - 213681227452 T + p^{15} T^{2} )( 1 + 213681227452 T + p^{15} T^{2} )$$
37$C_2$ $$( 1 + 1090158909950 T + p^{15} T^{2} )^{2}$$
41$C_2$ $$( 1 + p^{15} T^{2} )^{2}$$
43$C_2$ $$( 1 + 1440654152600 T + p^{15} T^{2} )^{2}$$
47$C_2$ $$( 1 + p^{15} T^{2} )^{2}$$
53$C_2$ $$( 1 - p^{15} T^{2} )^{2}$$
59$C_2$ $$( 1 + p^{15} T^{2} )^{2}$$
61$C_2$ $$( 1 - 40241378988902 T + p^{15} T^{2} )( 1 + 40241378988902 T + p^{15} T^{2} )$$
67$C_2$ $$( 1 - 99059017336400 T + p^{15} T^{2} )^{2}$$
71$C_2$ $$( 1 - p^{15} T^{2} )^{2}$$
73$C_2$ $$( 1 - 9014812804550 T + p^{15} T^{2} )( 1 + 9014812804550 T + p^{15} T^{2} )$$
79$C_2$ $$( 1 + 88692309079036 T + p^{15} T^{2} )^{2}$$
83$C_2$ $$( 1 + p^{15} T^{2} )^{2}$$
89$C_2$ $$( 1 + p^{15} T^{2} )^{2}$$
97$C_2$ $$( 1 - 1035097921427150 T + p^{15} T^{2} )( 1 + 1035097921427150 T + p^{15} T^{2} )$$
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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

−15.49324509119703693504009919440, −14.24082859765475427127382918343, −13.82914797863283444878026347711, −12.68591442736919373644606412592, −12.18309682557366742813087522021, −11.47215887844121575066050454796, −11.24357202339979074206706482467, −10.22921667753242064975918382585, −9.895183044066066667924107343752, −8.594137534394891621487304573129, −8.005847599001367050501114729258, −7.06305225098507092317833600335, −6.63712617479463320711074826739, −5.84155133974696674709532997182, −5.28425699911513668261171264844, −3.42968457279867998317305634211, −3.33454189204525733152853180207, −2.07261344498227344383184156871, −1.84438923898225316045784140169, −0.46919766004656336527176456339, 0.46919766004656336527176456339, 1.84438923898225316045784140169, 2.07261344498227344383184156871, 3.33454189204525733152853180207, 3.42968457279867998317305634211, 5.28425699911513668261171264844, 5.84155133974696674709532997182, 6.63712617479463320711074826739, 7.06305225098507092317833600335, 8.005847599001367050501114729258, 8.594137534394891621487304573129, 9.895183044066066667924107343752, 10.22921667753242064975918382585, 11.24357202339979074206706482467, 11.47215887844121575066050454796, 12.18309682557366742813087522021, 12.68591442736919373644606412592, 13.82914797863283444878026347711, 14.24082859765475427127382918343, 15.49324509119703693504009919440