Properties

Label 6-768e3-1.1-c3e3-0-3
Degree $6$
Conductor $452984832$
Sign $-1$
Analytic cond. $93042.6$
Root an. cond. $6.73152$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·3-s − 10·5-s − 14·7-s + 54·9-s − 52·13-s − 90·15-s + 26·17-s + 28·19-s − 126·21-s − 164·23-s − 111·25-s + 270·27-s − 174·29-s − 318·31-s + 140·35-s − 296·37-s − 468·39-s − 118·41-s − 260·43-s − 540·45-s − 204·47-s − 253·49-s + 234·51-s − 1.08e3·53-s + 252·57-s + 196·59-s − 1.53e3·61-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.894·5-s − 0.755·7-s + 2·9-s − 1.10·13-s − 1.54·15-s + 0.370·17-s + 0.338·19-s − 1.30·21-s − 1.48·23-s − 0.887·25-s + 1.92·27-s − 1.11·29-s − 1.84·31-s + 0.676·35-s − 1.31·37-s − 1.92·39-s − 0.449·41-s − 0.922·43-s − 1.78·45-s − 0.633·47-s − 0.737·49-s + 0.642·51-s − 2.81·53-s + 0.585·57-s + 0.432·59-s − 3.22·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{24} \cdot 3^{3}\)
Sign: $-1$
Analytic conductor: \(93042.6\)
Root analytic conductor: \(6.73152\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{24} \cdot 3^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ -1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - p T )^{3} \)
good5$S_4\times C_2$ \( 1 + 2 p T + 211 T^{2} + 2396 T^{3} + 211 p^{3} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} \)
7$S_4\times C_2$ \( 1 + 2 p T + 449 T^{2} + 3788 T^{3} + 449 p^{3} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 + 107 p T^{2} + 49152 T^{3} + 107 p^{4} T^{4} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 + 4 p T + 5487 T^{2} + 172616 T^{3} + 5487 p^{3} T^{4} + 4 p^{7} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 - 26 T + 3615 T^{2} + 222100 T^{3} + 3615 p^{3} T^{4} - 26 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 - 28 T + 9489 T^{2} - 658856 T^{3} + 9489 p^{3} T^{4} - 28 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 + 164 T + 42885 T^{2} + 4036280 T^{3} + 42885 p^{3} T^{4} + 164 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 + 6 p T + 77131 T^{2} + 8426004 T^{3} + 77131 p^{3} T^{4} + 6 p^{7} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 + 318 T + 93849 T^{2} + 15197452 T^{3} + 93849 p^{3} T^{4} + 318 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 + 8 p T + 105879 T^{2} + 29903632 T^{3} + 105879 p^{3} T^{4} + 8 p^{7} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 + 118 T + 89463 T^{2} - 3720620 T^{3} + 89463 p^{3} T^{4} + 118 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 + 260 T + 157545 T^{2} + 32742232 T^{3} + 157545 p^{3} T^{4} + 260 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 + 204 T + 283677 T^{2} + 40395048 T^{3} + 283677 p^{3} T^{4} + 204 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 + 1086 T + 736099 T^{2} + 344026068 T^{3} + 736099 p^{3} T^{4} + 1086 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 - 196 T + 566137 T^{2} - 71984984 T^{3} + 566137 p^{3} T^{4} - 196 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 + 1536 T + 1409583 T^{2} + 802126848 T^{3} + 1409583 p^{3} T^{4} + 1536 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 + 660 T + 592833 T^{2} + 364778232 T^{3} + 592833 p^{3} T^{4} + 660 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 - 12 p T + 1006773 T^{2} - 524795352 T^{3} + 1006773 p^{3} T^{4} - 12 p^{7} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 + 478 T + 911095 T^{2} + 251066948 T^{3} + 911095 p^{3} T^{4} + 478 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 + 22 T + 1407593 T^{2} + 13791100 T^{3} + 1407593 p^{3} T^{4} + 22 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 - 1136 T + 2100385 T^{2} - 1337053600 T^{3} + 2100385 p^{3} T^{4} - 1136 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 - 110 T + 2073543 T^{2} - 156516836 T^{3} + 2073543 p^{3} T^{4} - 110 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 + 1222 T + 2989679 T^{2} + 2155770388 T^{3} + 2989679 p^{3} T^{4} + 1222 p^{6} T^{5} + p^{9} T^{6} \)
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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

−9.384424845087646209869695248362, −9.068766849859078934598898127251, −8.463369908872854846108692571666, −8.443710305060498289394919412138, −7.952324061405263498256376759417, −7.74391613014132342933946137610, −7.69304314400839366408495973126, −7.36832651163906032613249984696, −7.01239861034029561043676956969, −6.78267625649447919227347726947, −6.17639915778606542546831788334, −6.15969501427863729453499039315, −5.68131497138935082017169115618, −5.01694501912531748581141289525, −4.99873764108689347763659360051, −4.63452589727187762109371054656, −3.98095284418687605348042933877, −3.71287558049740224150397939081, −3.68619244388086639367281149248, −3.28601969505736643145812901410, −2.83443316217434114884587196292, −2.61678448403935355973419816281, −1.81731388207632616679556938812, −1.74401850264026276100306910425, −1.44181291808602741287963678906, 0, 0, 0, 1.44181291808602741287963678906, 1.74401850264026276100306910425, 1.81731388207632616679556938812, 2.61678448403935355973419816281, 2.83443316217434114884587196292, 3.28601969505736643145812901410, 3.68619244388086639367281149248, 3.71287558049740224150397939081, 3.98095284418687605348042933877, 4.63452589727187762109371054656, 4.99873764108689347763659360051, 5.01694501912531748581141289525, 5.68131497138935082017169115618, 6.15969501427863729453499039315, 6.17639915778606542546831788334, 6.78267625649447919227347726947, 7.01239861034029561043676956969, 7.36832651163906032613249984696, 7.69304314400839366408495973126, 7.74391613014132342933946137610, 7.952324061405263498256376759417, 8.443710305060498289394919412138, 8.463369908872854846108692571666, 9.068766849859078934598898127251, 9.384424845087646209869695248362

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