# Properties

 Label 6-4840e3-1.1-c1e3-0-1 Degree $6$ Conductor $113379904000$ Sign $1$ Analytic cond. $57725.4$ Root an. cond. $6.21671$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3-s + 3·5-s − 7-s − 3·15-s + 7·17-s + 3·19-s + 21-s − 4·23-s + 6·25-s − 27-s + 5·29-s − 17·31-s − 3·35-s + 37-s − 2·41-s + 16·43-s − 6·47-s − 12·49-s − 7·51-s + 53-s − 3·57-s − 4·59-s + 11·61-s − 16·67-s + 4·69-s + 71-s − 22·73-s + ⋯
 L(s)  = 1 − 0.577·3-s + 1.34·5-s − 0.377·7-s − 0.774·15-s + 1.69·17-s + 0.688·19-s + 0.218·21-s − 0.834·23-s + 6/5·25-s − 0.192·27-s + 0.928·29-s − 3.05·31-s − 0.507·35-s + 0.164·37-s − 0.312·41-s + 2.43·43-s − 0.875·47-s − 1.71·49-s − 0.980·51-s + 0.137·53-s − 0.397·57-s − 0.520·59-s + 1.40·61-s − 1.95·67-s + 0.481·69-s + 0.118·71-s − 2.57·73-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$2^{9} \cdot 5^{3} \cdot 11^{6}$$ Sign: $1$ Analytic conductor: $$57725.4$$ Root analytic conductor: $$6.21671$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(6,\ 2^{9} \cdot 5^{3} \cdot 11^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$3.292938694$$ $$L(\frac12)$$ $$\approx$$ $$3.292938694$$ $$L(\frac{3}{2})$$ 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)$
bad2 $$1$$
5$C_1$ $$( 1 - T )^{3}$$
11 $$1$$
good3$S_4\times C_2$ $$1 + T + T^{2} + 2 T^{3} + p T^{4} + p^{2} T^{5} + p^{3} T^{6}$$
7$S_4\times C_2$ $$1 + T + 13 T^{2} + 10 T^{3} + 13 p T^{4} + p^{2} T^{5} + p^{3} T^{6}$$
13$S_4\times C_2$ $$1 - T^{2} + 64 T^{3} - p T^{4} + p^{3} T^{6}$$
17$S_4\times C_2$ $$1 - 7 T + 59 T^{2} - 230 T^{3} + 59 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6}$$
19$S_4\times C_2$ $$1 - 3 T + 17 T^{2} - 130 T^{3} + 17 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}$$
23$S_4\times C_2$ $$1 + 4 T + 49 T^{2} + 120 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
29$S_4\times C_2$ $$1 - 5 T + 71 T^{2} - 226 T^{3} + 71 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6}$$
31$S_4\times C_2$ $$1 + 17 T + 165 T^{2} + 1070 T^{3} + 165 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6}$$
37$S_4\times C_2$ $$1 - T + 87 T^{2} - 94 T^{3} + 87 p T^{4} - p^{2} T^{5} + p^{3} T^{6}$$
41$S_4\times C_2$ $$1 + 2 T + 91 T^{2} + 132 T^{3} + 91 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
43$S_4\times C_2$ $$1 - 16 T + 189 T^{2} - 1360 T^{3} + 189 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6}$$
47$S_4\times C_2$ $$1 + 6 T + 113 T^{2} + 556 T^{3} + 113 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}$$
53$S_4\times C_2$ $$1 - T + 135 T^{2} - 126 T^{3} + 135 p T^{4} - p^{2} T^{5} + p^{3} T^{6}$$
59$S_4\times C_2$ $$1 + 4 T + 81 T^{2} + 600 T^{3} + 81 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
61$S_4\times C_2$ $$1 - 11 T + 95 T^{2} - 6 p T^{3} + 95 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6}$$
67$S_4\times C_2$ $$1 + 16 T + 165 T^{2} + 1168 T^{3} + 165 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6}$$
71$S_4\times C_2$ $$1 - T + 85 T^{2} - 654 T^{3} + 85 p T^{4} - p^{2} T^{5} + p^{3} T^{6}$$
73$S_4\times C_2$ $$1 + 22 T + 355 T^{2} + 3388 T^{3} + 355 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6}$$
79$C_2$ $$( 1 - 12 T + p T^{2} )^{3}$$
83$S_4\times C_2$ $$1 + 21 T^{2} - 880 T^{3} + 21 p T^{4} + p^{3} T^{6}$$
89$S_4\times C_2$ $$1 + 3 T + 83 T^{2} + 1138 T^{3} + 83 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6}$$
97$C_2$ $$( 1 - 6 T + p T^{2} )^{3}$$
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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

−7.18844278970153240062772196954, −7.07062443922606054773999835769, −6.87222441427386248992585788966, −6.56893653642224372216609010411, −6.09856857895543836720010814584, −6.02753710279672125882058815533, −5.99025267337360975318253354586, −5.53555113740852080194256908235, −5.40432573085708568751504697625, −5.33004791085783188706903468343, −4.92950860504219057940900747911, −4.64505150065000630832444124689, −4.38530430470477934204038453786, −3.91599166394901374814750864615, −3.64711951760901079963921666733, −3.63053328764205418069762730188, −2.94267794980687641536668160063, −2.87231592728251651165437308048, −2.82349758898379213675780626550, −1.99560729488448757461083600834, −1.87584195840494305369279306852, −1.67770646336179873720831323873, −1.23595057029947059887455714779, −0.74558954190318923600909187921, −0.38042757493823765793182904637, 0.38042757493823765793182904637, 0.74558954190318923600909187921, 1.23595057029947059887455714779, 1.67770646336179873720831323873, 1.87584195840494305369279306852, 1.99560729488448757461083600834, 2.82349758898379213675780626550, 2.87231592728251651165437308048, 2.94267794980687641536668160063, 3.63053328764205418069762730188, 3.64711951760901079963921666733, 3.91599166394901374814750864615, 4.38530430470477934204038453786, 4.64505150065000630832444124689, 4.92950860504219057940900747911, 5.33004791085783188706903468343, 5.40432573085708568751504697625, 5.53555113740852080194256908235, 5.99025267337360975318253354586, 6.02753710279672125882058815533, 6.09856857895543836720010814584, 6.56893653642224372216609010411, 6.87222441427386248992585788966, 7.07062443922606054773999835769, 7.18844278970153240062772196954