Properties

Label 6-1815e3-1.1-c3e3-0-3
Degree $6$
Conductor $5979018375$
Sign $-1$
Analytic cond. $1.22808\times 10^{6}$
Root an. cond. $10.3483$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 9·3-s + 7·4-s − 15·5-s − 36·6-s + 4·7-s + 12·8-s + 54·9-s − 60·10-s − 63·12-s + 16·14-s + 135·15-s − 7·16-s + 218·17-s + 216·18-s − 146·19-s − 105·20-s − 36·21-s − 200·23-s − 108·24-s + 150·25-s − 270·27-s + 28·28-s − 68·29-s + 540·30-s − 68·31-s − 28·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.73·3-s + 7/8·4-s − 1.34·5-s − 2.44·6-s + 0.215·7-s + 0.530·8-s + 2·9-s − 1.89·10-s − 1.51·12-s + 0.305·14-s + 2.32·15-s − 0.109·16-s + 3.11·17-s + 2.82·18-s − 1.76·19-s − 1.17·20-s − 0.374·21-s − 1.81·23-s − 0.918·24-s + 6/5·25-s − 1.92·27-s + 0.188·28-s − 0.435·29-s + 3.28·30-s − 0.393·31-s − 0.154·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{3} \cdot 5^{3} \cdot 11^{6}\)
Sign: $-1$
Analytic conductor: \(1.22808\times 10^{6}\)
Root analytic conductor: \(10.3483\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 3^{3} \cdot 5^{3} \cdot 11^{6} ,\ ( \ : 3/2, 3/2, 3/2 ),\ -1 )\)

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)$
bad3$C_1$ \( ( 1 + p T )^{3} \)
5$C_1$ \( ( 1 + p T )^{3} \)
11 \( 1 \)
good2$S_4\times C_2$ \( 1 - p^{2} T + 9 T^{2} - 5 p^{2} T^{3} + 9 p^{3} T^{4} - p^{8} T^{5} + p^{9} T^{6} \)
7$S_4\times C_2$ \( 1 - 4 T + 601 T^{2} + 1000 T^{3} + 601 p^{3} T^{4} - 4 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 + 3031 T^{2} + 34144 T^{3} + 3031 p^{3} T^{4} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 - 218 T + 24419 T^{2} - 1906964 T^{3} + 24419 p^{3} T^{4} - 218 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 + 146 T + 25953 T^{2} + 2033788 T^{3} + 25953 p^{3} T^{4} + 146 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 + 200 T + 33829 T^{2} + 4868464 T^{3} + 33829 p^{3} T^{4} + 200 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 + 68 T + 18803 T^{2} + 153848 T^{3} + 18803 p^{3} T^{4} + 68 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 + 68 T + 66141 T^{2} + 2239480 T^{3} + 66141 p^{3} T^{4} + 68 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 + 390 T + 188419 T^{2} + 40128292 T^{3} + 188419 p^{3} T^{4} + 390 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 - 196 T + 4407 p T^{2} - 22652824 T^{3} + 4407 p^{4} T^{4} - 196 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 - 524 T + 209853 T^{2} - 52049416 T^{3} + 209853 p^{3} T^{4} - 524 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 + 60 T + 175549 T^{2} - 8508216 T^{3} + 175549 p^{3} T^{4} + 60 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 + 158 T + 185779 T^{2} + 86620084 T^{3} + 185779 p^{3} T^{4} + 158 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 + 1044 T + 680281 T^{2} + 344604088 T^{3} + 680281 p^{3} T^{4} + 1044 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 + 642 T + 620395 T^{2} + 268686220 T^{3} + 620395 p^{3} T^{4} + 642 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 + 236 T + 440081 T^{2} + 229497800 T^{3} + 440081 p^{3} T^{4} + 236 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 + 544 T + 944005 T^{2} + 395960768 T^{3} + 944005 p^{3} T^{4} + 544 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 + 900 T + 867475 T^{2} + 705839944 T^{3} + 867475 p^{3} T^{4} + 900 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 - 1586 T + 2041325 T^{2} - 1549224716 T^{3} + 2041325 p^{3} T^{4} - 1586 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 - 1582 T + 1407717 T^{2} - 884488684 T^{3} + 1407717 p^{3} T^{4} - 1582 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 + 2122 T + 3521847 T^{2} + 3285333068 T^{3} + 3521847 p^{3} T^{4} + 2122 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 - 618 T + 1446319 T^{2} - 1351607564 T^{3} + 1446319 p^{3} T^{4} - 618 p^{6} T^{5} + p^{9} T^{6} \)
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

−8.072754521640923200279475969157, −7.76481259739063057927867041688, −7.65252622687174115385609345788, −7.40711684547733782273394111388, −7.35863018972085559916540332048, −6.67979398888853957502572180066, −6.51397779865814228072301631413, −6.21092342641566896105627523170, −6.10969627197443293184321333357, −5.81147372032933754121635651047, −5.34509162263331408617604789477, −5.19038654321030509811003287300, −5.18680411328066970086039702876, −4.43787823244643663199236705147, −4.40638030267282377004497718832, −4.31009615260131408553926853374, −3.93830525393156702777458907153, −3.47423310069009894467754529805, −3.44169486757392661055203685956, −2.97992353552746200113573371552, −2.55309200048611741905301304638, −1.82971697880640386232104031218, −1.78205462058211563833813816906, −1.11443549351482892981802612504, −1.02438152659599502048358527167, 0, 0, 0, 1.02438152659599502048358527167, 1.11443549351482892981802612504, 1.78205462058211563833813816906, 1.82971697880640386232104031218, 2.55309200048611741905301304638, 2.97992353552746200113573371552, 3.44169486757392661055203685956, 3.47423310069009894467754529805, 3.93830525393156702777458907153, 4.31009615260131408553926853374, 4.40638030267282377004497718832, 4.43787823244643663199236705147, 5.18680411328066970086039702876, 5.19038654321030509811003287300, 5.34509162263331408617604789477, 5.81147372032933754121635651047, 6.10969627197443293184321333357, 6.21092342641566896105627523170, 6.51397779865814228072301631413, 6.67979398888853957502572180066, 7.35863018972085559916540332048, 7.40711684547733782273394111388, 7.65252622687174115385609345788, 7.76481259739063057927867041688, 8.072754521640923200279475969157

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