# Properties

 Label 6-825e3-1.1-c5e3-0-4 Degree $6$ Conductor $561515625$ Sign $-1$ Analytic cond. $2.31655\times 10^{6}$ Root an. cond. $11.5028$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $3$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s + 27·3-s − 67·4-s − 54·6-s + 68·7-s + 166·8-s + 486·9-s + 363·11-s − 1.80e3·12-s − 290·13-s − 136·14-s + 2.37e3·16-s − 434·17-s − 972·18-s − 2.85e3·19-s + 1.83e3·21-s − 726·22-s + 640·23-s + 4.48e3·24-s + 580·26-s + 7.29e3·27-s − 4.55e3·28-s − 4.53e3·29-s − 1.49e4·31-s − 6.66e3·32-s + 9.80e3·33-s + 868·34-s + ⋯
 L(s)  = 1 − 0.353·2-s + 1.73·3-s − 2.09·4-s − 0.612·6-s + 0.524·7-s + 0.917·8-s + 2·9-s + 0.904·11-s − 3.62·12-s − 0.475·13-s − 0.185·14-s + 2.31·16-s − 0.364·17-s − 0.707·18-s − 1.81·19-s + 0.908·21-s − 0.319·22-s + 0.252·23-s + 1.58·24-s + 0.168·26-s + 1.92·27-s − 1.09·28-s − 1.00·29-s − 2.79·31-s − 1.15·32-s + 1.56·33-s + 0.128·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(6-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$3^{3} \cdot 5^{6} \cdot 11^{3}$$ Sign: $-1$ Analytic conductor: $$2.31655\times 10^{6}$$ Root analytic conductor: $$11.5028$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$3$$ Selberg data: $$(6,\ 3^{3} \cdot 5^{6} \cdot 11^{3} ,\ ( \ : 5/2, 5/2, 5/2 ),\ -1 )$$

## Particular Values

 $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ 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$C_1$ $$( 1 - p^{2} T )^{3}$$
5 $$1$$
11$C_1$ $$( 1 - p^{2} T )^{3}$$
good2$S_4\times C_2$ $$1 + p T + 71 T^{2} + 55 p T^{3} + 71 p^{5} T^{4} + p^{11} T^{5} + p^{15} T^{6}$$
7$S_4\times C_2$ $$1 - 68 T + 34869 T^{2} - 2533560 T^{3} + 34869 p^{5} T^{4} - 68 p^{10} T^{5} + p^{15} T^{6}$$
13$S_4\times C_2$ $$1 + 290 T + 658819 T^{2} + 83286348 T^{3} + 658819 p^{5} T^{4} + 290 p^{10} T^{5} + p^{15} T^{6}$$
17$S_4\times C_2$ $$1 + 434 T + 58991 T^{2} - 1314616612 T^{3} + 58991 p^{5} T^{4} + 434 p^{10} T^{5} + p^{15} T^{6}$$
19$S_4\times C_2$ $$1 + 2856 T + 8753113 T^{2} + 14281181168 T^{3} + 8753113 p^{5} T^{4} + 2856 p^{10} T^{5} + p^{15} T^{6}$$
23$S_4\times C_2$ $$1 - 640 T + 6700261 T^{2} + 7538830592 T^{3} + 6700261 p^{5} T^{4} - 640 p^{10} T^{5} + p^{15} T^{6}$$
29$S_4\times C_2$ $$1 + 4538 T + 35092643 T^{2} + 141745639868 T^{3} + 35092643 p^{5} T^{4} + 4538 p^{10} T^{5} + p^{15} T^{6}$$
31$S_4\times C_2$ $$1 + 14968 T + 160071261 T^{2} + 978687787280 T^{3} + 160071261 p^{5} T^{4} + 14968 p^{10} T^{5} + p^{15} T^{6}$$
37$S_4\times C_2$ $$1 - 6190 T + 113089771 T^{2} - 886491849396 T^{3} + 113089771 p^{5} T^{4} - 6190 p^{10} T^{5} + p^{15} T^{6}$$
41$S_4\times C_2$ $$1 + 8926 T + 272145047 T^{2} + 2187570068644 T^{3} + 272145047 p^{5} T^{4} + 8926 p^{10} T^{5} + p^{15} T^{6}$$
43$S_4\times C_2$ $$1 - 33592 T + 693464257 T^{2} - 9837617686992 T^{3} + 693464257 p^{5} T^{4} - 33592 p^{10} T^{5} + p^{15} T^{6}$$
47$S_4\times C_2$ $$1 - 24640 T + 788101261 T^{2} - 10622124840832 T^{3} + 788101261 p^{5} T^{4} - 24640 p^{10} T^{5} + p^{15} T^{6}$$
53$S_4\times C_2$ $$1 - 22934 T + 1099334651 T^{2} - 14788031800868 T^{3} + 1099334651 p^{5} T^{4} - 22934 p^{10} T^{5} + p^{15} T^{6}$$
59$S_4\times C_2$ $$1 + 13756 T - 129165359 T^{2} - 1129007325848 T^{3} - 129165359 p^{5} T^{4} + 13756 p^{10} T^{5} + p^{15} T^{6}$$
61$S_4\times C_2$ $$1 - 24602 T + 2638007155 T^{2} - 41514451599900 T^{3} + 2638007155 p^{5} T^{4} - 24602 p^{10} T^{5} + p^{15} T^{6}$$
67$S_4\times C_2$ $$1 + 16868 T + 3843672649 T^{2} + 43721067889560 T^{3} + 3843672649 p^{5} T^{4} + 16868 p^{10} T^{5} + p^{15} T^{6}$$
71$S_4\times C_2$ $$1 - 4856 T + 1951490165 T^{2} + 16632390591088 T^{3} + 1951490165 p^{5} T^{4} - 4856 p^{10} T^{5} + p^{15} T^{6}$$
73$S_4\times C_2$ $$1 + 1910 T + 5875530055 T^{2} + 8082237219348 T^{3} + 5875530055 p^{5} T^{4} + 1910 p^{10} T^{5} + p^{15} T^{6}$$
79$S_4\times C_2$ $$1 + 36844 T + 4302241965 T^{2} + 155969878390184 T^{3} + 4302241965 p^{5} T^{4} + 36844 p^{10} T^{5} + p^{15} T^{6}$$
83$S_4\times C_2$ $$1 - 48796 T + 9666032953 T^{2} - 314032777245928 T^{3} + 9666032953 p^{5} T^{4} - 48796 p^{10} T^{5} + p^{15} T^{6}$$
89$S_4\times C_2$ $$1 + 188978 T + 25306287767 T^{2} + 2094440123430812 T^{3} + 25306287767 p^{5} T^{4} + 188978 p^{10} T^{5} + p^{15} T^{6}$$
97$S_4\times C_2$ $$1 + 247526 T + 41254986031 T^{2} + 4400049090000852 T^{3} + 41254986031 p^{5} T^{4} + 247526 p^{10} T^{5} + p^{15} T^{6}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

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

−8.867436054936391435373700549880, −8.428959922781021647355991791509, −8.336691385767081788545429421265, −8.323082667941505028636600858187, −7.52952586048962628361150951615, −7.47924671998887287775658630026, −7.42310795034946919358928784140, −6.91401248555190957101366837079, −6.46062820198606362844246076825, −6.26873064167609406957357295898, −5.52730126559955464673100110921, −5.48504101154572855845718291019, −5.23806053301703231394592305812, −4.48601481583481356968311315583, −4.48322476342371146097257545787, −4.20850074338068593651839012484, −3.80315682864487268670107460561, −3.77849809297233710936134131110, −3.41723391003108686861821456561, −2.55672510736687406640517922920, −2.49975657489757273391537176730, −2.19454320605859059652973046126, −1.55542013325244813116500576175, −1.16379482382485122669916559587, −1.14705849442005581393992145520, 0, 0, 0, 1.14705849442005581393992145520, 1.16379482382485122669916559587, 1.55542013325244813116500576175, 2.19454320605859059652973046126, 2.49975657489757273391537176730, 2.55672510736687406640517922920, 3.41723391003108686861821456561, 3.77849809297233710936134131110, 3.80315682864487268670107460561, 4.20850074338068593651839012484, 4.48322476342371146097257545787, 4.48601481583481356968311315583, 5.23806053301703231394592305812, 5.48504101154572855845718291019, 5.52730126559955464673100110921, 6.26873064167609406957357295898, 6.46062820198606362844246076825, 6.91401248555190957101366837079, 7.42310795034946919358928784140, 7.47924671998887287775658630026, 7.52952586048962628361150951615, 8.323082667941505028636600858187, 8.336691385767081788545429421265, 8.428959922781021647355991791509, 8.867436054936391435373700549880