Properties

Label 6-2200e3-1.1-c1e3-0-0
Degree $6$
Conductor $10648000000$
Sign $1$
Analytic cond. $5421.24$
Root an. cond. $4.19131$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·7-s − 9-s + 3·11-s − 3·13-s + 5·17-s + 7·19-s + 3·21-s − 10·23-s + 3·27-s + 10·29-s − 3·31-s − 3·33-s + 8·37-s + 3·39-s + 4·41-s − 9·43-s − 49-s − 5·51-s − 5·53-s − 7·57-s − 61-s + 3·63-s + 14·67-s + 10·69-s + 17·71-s + 21·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.13·7-s − 1/3·9-s + 0.904·11-s − 0.832·13-s + 1.21·17-s + 1.60·19-s + 0.654·21-s − 2.08·23-s + 0.577·27-s + 1.85·29-s − 0.538·31-s − 0.522·33-s + 1.31·37-s + 0.480·39-s + 0.624·41-s − 1.37·43-s − 1/7·49-s − 0.700·51-s − 0.686·53-s − 0.927·57-s − 0.128·61-s + 0.377·63-s + 1.71·67-s + 1.20·69-s + 2.01·71-s + 2.45·73-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 5^{6} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(5421.24\)
Root analytic conductor: \(4.19131\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 5^{6} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.036647192\)
\(L(\frac12)\) \(\approx\) \(2.036647192\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_1$ \( ( 1 - T )^{3} \)
good3$S_4\times C_2$ \( 1 + T + 2 T^{2} + 2 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + 3 T + 10 T^{2} + 10 T^{3} + 10 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{3} \)
17$S_4\times C_2$ \( 1 - 5 T + 52 T^{2} - 166 T^{3} + 52 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 7 T + 44 T^{2} - 239 T^{3} + 44 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 10 T + 95 T^{2} + 469 T^{3} + 95 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 10 T + 79 T^{2} - 379 T^{3} + 79 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 3 T + 40 T^{2} - 21 T^{3} + 40 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 8 T + 110 T^{2} - 548 T^{3} + 110 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 4 T + 106 T^{2} - 304 T^{3} + 106 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 9 T + 73 T^{2} + 266 T^{3} + 73 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 64 T^{2} - 192 T^{3} + 64 p T^{4} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 5 T + 18 T^{2} + 388 T^{3} + 18 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 100 T^{2} + 192 T^{3} + 100 p T^{4} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + T + 142 T^{2} + 8 T^{3} + 142 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 14 T + 185 T^{2} - 1780 T^{3} + 185 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 17 T + 287 T^{2} - 2466 T^{3} + 287 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 21 T + 240 T^{2} - 2014 T^{3} + 240 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 11 T + 216 T^{2} - 1594 T^{3} + 216 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 20 T + 255 T^{2} + 2491 T^{3} + 255 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 30 T + 553 T^{2} - 6219 T^{3} + 553 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 10 T + 259 T^{2} - 1917 T^{3} + 259 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.211459022331313316708857322891, −7.64792962566346164825674541541, −7.63984965333184882029925758030, −7.16299337478275751075407597677, −6.95873998451486382912092310278, −6.60468059558720731752068743228, −6.51617159201195940170550327891, −6.01683148790183519477888003266, −5.99105333677367314401071032042, −5.87041131645338975532760063743, −5.27619642434126653078725150849, −5.02429193496024749111496118442, −4.98767223769086838781376570756, −4.45372340377730391920079787972, −4.27721370510737966900077545707, −3.73733020026033455324557056640, −3.44993829169380887729889100490, −3.23544908485425440822844515493, −3.20000392612341435952889466879, −2.50564410769329498266939232607, −2.10938615306349937776964465552, −2.00810374247608213466951334310, −1.11458365273105920021744963561, −0.874525955513996168433282289909, −0.44086114792754937610326039081, 0.44086114792754937610326039081, 0.874525955513996168433282289909, 1.11458365273105920021744963561, 2.00810374247608213466951334310, 2.10938615306349937776964465552, 2.50564410769329498266939232607, 3.20000392612341435952889466879, 3.23544908485425440822844515493, 3.44993829169380887729889100490, 3.73733020026033455324557056640, 4.27721370510737966900077545707, 4.45372340377730391920079787972, 4.98767223769086838781376570756, 5.02429193496024749111496118442, 5.27619642434126653078725150849, 5.87041131645338975532760063743, 5.99105333677367314401071032042, 6.01683148790183519477888003266, 6.51617159201195940170550327891, 6.60468059558720731752068743228, 6.95873998451486382912092310278, 7.16299337478275751075407597677, 7.63984965333184882029925758030, 7.64792962566346164825674541541, 8.211459022331313316708857322891

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