Properties

Label 6-8470e3-1.1-c1e3-0-4
Degree $6$
Conductor $607645423000$
Sign $1$
Analytic cond. $309372.$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 6·4-s − 3·5-s + 3·7-s + 10·8-s + 9-s − 9·10-s + 9·14-s + 15·16-s + 8·17-s + 3·18-s + 2·19-s − 18·20-s + 2·23-s + 6·25-s + 4·27-s + 18·28-s − 4·29-s + 18·31-s + 21·32-s + 24·34-s − 9·35-s + 6·36-s + 4·37-s + 6·38-s − 30·40-s + 18·41-s + ⋯
L(s)  = 1  + 2.12·2-s + 3·4-s − 1.34·5-s + 1.13·7-s + 3.53·8-s + 1/3·9-s − 2.84·10-s + 2.40·14-s + 15/4·16-s + 1.94·17-s + 0.707·18-s + 0.458·19-s − 4.02·20-s + 0.417·23-s + 6/5·25-s + 0.769·27-s + 3.40·28-s − 0.742·29-s + 3.23·31-s + 3.71·32-s + 4.11·34-s − 1.52·35-s + 36-s + 0.657·37-s + 0.973·38-s − 4.74·40-s + 2.81·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(309372.\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(37.68056202\)
\(L(\frac12)\) \(\approx\) \(37.68056202\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
5$C_1$ \( ( 1 + T )^{3} \)
7$C_1$ \( ( 1 - T )^{3} \)
11 \( 1 \)
good3$S_4\times C_2$ \( 1 - T^{2} - 4 T^{3} - p T^{4} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - T^{2} + 32 T^{3} - p T^{4} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 8 T + 39 T^{2} - 144 T^{3} + 39 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 2 T + 27 T^{2} - 20 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 2 T + 39 T^{2} - 36 T^{3} + 39 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 4 T + 5 T^{2} - 196 T^{3} + 5 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{3} \)
37$S_4\times C_2$ \( 1 - 4 T + 61 T^{2} - 172 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 18 T + 221 T^{2} - 1636 T^{3} + 221 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 8 T + 101 T^{2} + 672 T^{3} + 101 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 2 T + 93 T^{2} - 252 T^{3} + 93 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 4 T + 77 T^{2} + 4 T^{3} + 77 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 10 T + 113 T^{2} - 572 T^{3} + 113 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 20 T + 267 T^{2} + 2504 T^{3} + 267 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{3} \)
71$S_4\times C_2$ \( 1 - 8 T + 109 T^{2} - 1104 T^{3} + 109 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 4 T + 79 T^{2} + 200 T^{3} + 79 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 6 T - 25 T^{2} + 1324 T^{3} - 25 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 24 T + 401 T^{2} - 4144 T^{3} + 401 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 6 T + 239 T^{2} + 1028 T^{3} + 239 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 8 T + 281 T^{2} - 1452 T^{3} + 281 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
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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

−6.77490887590410750137837855785, −6.70114565029801410788447844128, −6.28461714961523221721868229268, −6.25822424405428322403524416026, −5.77878172686810871034776302648, −5.68155524975502394972783519196, −5.39597517880858293851267720083, −4.99209107510900734276816941065, −4.93908708596876869749480126943, −4.85320203704240796819945573233, −4.58392297523145711741684955812, −4.21755423131753341217116588301, −4.05378697989220204678887863684, −3.74106618821959260752382278174, −3.62606961683608995099151634613, −3.42450665701760012607959869511, −2.95372516871082483396428438223, −2.78346508447231219363337565768, −2.58612239454549606812151712882, −2.16218985638327320745750435997, −2.08269360183135597600494968686, −1.22901559704816844701658836459, −1.15680494216083921440425581694, −0.975443335722588104259528241418, −0.61200122363716809953341445245, 0.61200122363716809953341445245, 0.975443335722588104259528241418, 1.15680494216083921440425581694, 1.22901559704816844701658836459, 2.08269360183135597600494968686, 2.16218985638327320745750435997, 2.58612239454549606812151712882, 2.78346508447231219363337565768, 2.95372516871082483396428438223, 3.42450665701760012607959869511, 3.62606961683608995099151634613, 3.74106618821959260752382278174, 4.05378697989220204678887863684, 4.21755423131753341217116588301, 4.58392297523145711741684955812, 4.85320203704240796819945573233, 4.93908708596876869749480126943, 4.99209107510900734276816941065, 5.39597517880858293851267720083, 5.68155524975502394972783519196, 5.77878172686810871034776302648, 6.25822424405428322403524416026, 6.28461714961523221721868229268, 6.70114565029801410788447844128, 6.77490887590410750137837855785

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