Properties

Label 6-3800e3-1.1-c1e3-0-1
Degree $6$
Conductor $54872000000$
Sign $1$
Analytic cond. $27937.1$
Root an. cond. $5.50846$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s − 2·9-s − 5·13-s − 5·17-s + 3·19-s + 21-s + 23-s − 3·27-s + 17·29-s + 2·31-s − 8·37-s − 5·39-s + 6·41-s + 10·43-s − 4·47-s − 8·49-s − 5·51-s − 5·53-s + 3·57-s + 15·59-s + 8·61-s − 2·63-s − 3·67-s + 69-s − 4·71-s − 3·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s − 2/3·9-s − 1.38·13-s − 1.21·17-s + 0.688·19-s + 0.218·21-s + 0.208·23-s − 0.577·27-s + 3.15·29-s + 0.359·31-s − 1.31·37-s − 0.800·39-s + 0.937·41-s + 1.52·43-s − 0.583·47-s − 8/7·49-s − 0.700·51-s − 0.686·53-s + 0.397·57-s + 1.95·59-s + 1.02·61-s − 0.251·63-s − 0.366·67-s + 0.120·69-s − 0.474·71-s − 0.351·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 5^{6} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(27937.1\)
Root analytic conductor: \(5.50846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.461517656\)
\(L(\frac12)\) \(\approx\) \(4.461517656\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 - T )^{3} \)
good3$S_4\times C_2$ \( 1 - T + p T^{2} - 2 T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - T + 9 T^{2} + 2 T^{3} + 9 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
11$C_2$ \( ( 1 + p T^{2} )^{3} \)
13$S_4\times C_2$ \( 1 + 5 T + 41 T^{2} + 126 T^{3} + 41 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 5 T + 47 T^{2} + 166 T^{3} + 47 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - T + 37 T^{2} + 18 T^{3} + 37 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 17 T + 171 T^{2} - 1110 T^{3} + 171 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 2 T + 45 T^{2} + 4 T^{3} + 45 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 8 T + 3 p T^{2} + 584 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 6 T + 71 T^{2} - 436 T^{3} + 71 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 10 T + 137 T^{2} - 828 T^{3} + 137 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 4 T + 61 T^{2} + 504 T^{3} + 61 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 5 T + 117 T^{2} + 582 T^{3} + 117 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 15 T + 149 T^{2} - 986 T^{3} + 149 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 8 T + 179 T^{2} - 912 T^{3} + 179 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 3 T + 23 T^{2} - 650 T^{3} + 23 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 4 T + 21 T^{2} - 456 T^{3} + 21 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 3 T + 119 T^{2} + 10 p T^{3} + 119 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 2 T + 213 T^{2} - 284 T^{3} + 213 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 10 T + 233 T^{2} - 1628 T^{3} + 233 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 6 T + 215 T^{2} - 884 T^{3} + 215 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 4 T + 187 T^{2} - 1072 T^{3} + 187 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.66971334114824390960454798374, −7.32473634266634908536826590450, −6.92006838893277712040393586008, −6.91571282832885929961100313544, −6.47112873718190363658502906771, −6.39170200229673746009441480516, −6.18722576639986749341417432691, −5.60219585533581169831376611660, −5.50277231845999237152382609342, −5.27366477963395829838655696950, −4.80589408029310738882199073397, −4.67309323688724062202342399518, −4.62540801797066833083248161955, −4.24965189345675049307551633763, −3.71262750521123271427150898397, −3.71062852587024621764288342608, −2.96081568062518995074100532287, −2.90132907773152857585560505923, −2.89328860759419983623184038145, −2.33479467835769247911166676267, −2.07285257420737636423374911168, −1.83059318664171725861266230789, −1.23386602588461115297009009683, −0.60332134381119323186284653228, −0.56704580160764500942140021190, 0.56704580160764500942140021190, 0.60332134381119323186284653228, 1.23386602588461115297009009683, 1.83059318664171725861266230789, 2.07285257420737636423374911168, 2.33479467835769247911166676267, 2.89328860759419983623184038145, 2.90132907773152857585560505923, 2.96081568062518995074100532287, 3.71062852587024621764288342608, 3.71262750521123271427150898397, 4.24965189345675049307551633763, 4.62540801797066833083248161955, 4.67309323688724062202342399518, 4.80589408029310738882199073397, 5.27366477963395829838655696950, 5.50277231845999237152382609342, 5.60219585533581169831376611660, 6.18722576639986749341417432691, 6.39170200229673746009441480516, 6.47112873718190363658502906771, 6.91571282832885929961100313544, 6.92006838893277712040393586008, 7.32473634266634908536826590450, 7.66971334114824390960454798374

Graph of the $Z$-function along the critical line