Properties

Label 6-1400e3-1.1-c3e3-0-0
Degree $6$
Conductor $2744000000$
Sign $1$
Analytic cond. $563614.$
Root an. cond. $9.08860$
Motivic weight $3$
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  + 6·3-s + 21·7-s − 18·9-s − 54·11-s + 24·13-s − 48·17-s − 132·19-s + 126·21-s + 144·23-s − 136·27-s − 12·29-s − 180·31-s − 324·33-s + 402·37-s + 144·39-s − 66·41-s + 684·43-s + 1.07e3·47-s + 294·49-s − 288·51-s − 126·53-s − 792·57-s − 36·59-s − 750·61-s − 378·63-s + 1.23e3·67-s + 864·69-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.13·7-s − 2/3·9-s − 1.48·11-s + 0.512·13-s − 0.684·17-s − 1.59·19-s + 1.30·21-s + 1.30·23-s − 0.969·27-s − 0.0768·29-s − 1.04·31-s − 1.70·33-s + 1.78·37-s + 0.591·39-s − 0.251·41-s + 2.42·43-s + 3.33·47-s + 6/7·49-s − 0.790·51-s − 0.326·53-s − 1.84·57-s − 0.0794·59-s − 1.57·61-s − 0.755·63-s + 2.25·67-s + 1.50·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 5^{6} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(563614.\)
Root analytic conductor: \(9.08860\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 5^{6} \cdot 7^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(7.815290643\)
\(L(\frac12)\) \(\approx\) \(7.815290643\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - p T )^{3} \)
good3$S_4\times C_2$ \( 1 - 2 p T + 2 p^{3} T^{2} - 296 T^{3} + 2 p^{6} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 + 54 T + 270 p T^{2} + 1168 p^{2} T^{3} + 270 p^{4} T^{4} + 54 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 - 24 T - 504 T^{2} - 25938 T^{3} - 504 p^{3} T^{4} - 24 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 + 48 T + 60 T^{2} - 487478 T^{3} + 60 p^{3} T^{4} + 48 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 + 132 T + 18957 T^{2} + 1485128 T^{3} + 18957 p^{3} T^{4} + 132 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 - 144 T + 21681 T^{2} - 2637632 T^{3} + 21681 p^{3} T^{4} - 144 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 + 12 T + 59124 T^{2} + 302802 T^{3} + 59124 p^{3} T^{4} + 12 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 + 180 T + 92781 T^{2} + 10438040 T^{3} + 92781 p^{3} T^{4} + 180 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 - 402 T + 151395 T^{2} - 35883884 T^{3} + 151395 p^{3} T^{4} - 402 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 + 66 T + 71427 T^{2} + 20961284 T^{3} + 71427 p^{3} T^{4} + 66 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 - 684 T + 186309 T^{2} - 36930264 T^{3} + 186309 p^{3} T^{4} - 684 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 - 1074 T + 630762 T^{2} - 245857116 T^{3} + 630762 p^{3} T^{4} - 1074 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 + 126 T + 292167 T^{2} + 54269516 T^{3} + 292167 p^{3} T^{4} + 126 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 + 36 T + 240249 T^{2} + 340808 p T^{3} + 240249 p^{3} T^{4} + 36 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 + 750 T + 562599 T^{2} + 334891308 T^{3} + 562599 p^{3} T^{4} + 750 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 - 1236 T + 1279761 T^{2} - 750429880 T^{3} + 1279761 p^{3} T^{4} - 1236 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 + 696 T + 744453 T^{2} + 525131536 T^{3} + 744453 p^{3} T^{4} + 696 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 + 654 T + 1249383 T^{2} + 505892676 T^{3} + 1249383 p^{3} T^{4} + 654 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 - 762 T + 487446 T^{2} - 333544412 T^{3} + 487446 p^{3} T^{4} - 762 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 + 444 T + 1708113 T^{2} + 500309864 T^{3} + 1708113 p^{3} T^{4} + 444 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 - 1782 T + 1936227 T^{2} - 1676216940 T^{3} + 1936227 p^{3} T^{4} - 1782 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 + 1560 T + 2682612 T^{2} + 2844258426 T^{3} + 2682612 p^{3} T^{4} + 1560 p^{6} T^{5} + p^{9} T^{6} \)
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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

−8.334123817668410849958505043888, −7.76884473069054336360609560784, −7.64579808900323653214022345058, −7.57856078235931548703350527377, −7.15801567272949666364833357227, −6.87813550852377931912811541125, −6.42956699345020472114428948329, −5.97305743802992856255659647123, −5.95757905044937270644459579090, −5.68965956432073648465374384738, −5.14727343332808062791428594178, −5.10813637137072954881667067611, −4.65814559125413677671910381391, −4.22848804912579849776865627914, −4.06433234488837905643653480443, −3.95148821031350852551596811436, −3.00635769631397325002915386962, −2.95864728282863273392358095661, −2.91497678036121429846787507140, −2.17606023547636950709084300766, −2.14392979498255129823127291826, −1.93520355037949493853013241133, −1.07190993181509909816479102620, −0.62609808593242834172346109532, −0.48060604027253525022840684916, 0.48060604027253525022840684916, 0.62609808593242834172346109532, 1.07190993181509909816479102620, 1.93520355037949493853013241133, 2.14392979498255129823127291826, 2.17606023547636950709084300766, 2.91497678036121429846787507140, 2.95864728282863273392358095661, 3.00635769631397325002915386962, 3.95148821031350852551596811436, 4.06433234488837905643653480443, 4.22848804912579849776865627914, 4.65814559125413677671910381391, 5.10813637137072954881667067611, 5.14727343332808062791428594178, 5.68965956432073648465374384738, 5.95757905044937270644459579090, 5.97305743802992856255659647123, 6.42956699345020472114428948329, 6.87813550852377931912811541125, 7.15801567272949666364833357227, 7.57856078235931548703350527377, 7.64579808900323653214022345058, 7.76884473069054336360609560784, 8.334123817668410849958505043888

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