Properties

Degree 6
Conductor $ 3^{3} \cdot 7^{6} \cdot 41^{3} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 3

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3·3-s − 4-s − 4·5-s + 3·6-s − 8-s + 6·9-s − 4·10-s − 4·11-s − 3·12-s − 8·13-s − 12·15-s − 16-s − 2·17-s + 6·18-s − 2·19-s + 4·20-s − 4·22-s − 10·23-s − 3·24-s + 3·25-s − 8·26-s + 10·27-s − 6·29-s − 12·30-s + 2·31-s − 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s − 1/2·4-s − 1.78·5-s + 1.22·6-s − 0.353·8-s + 2·9-s − 1.26·10-s − 1.20·11-s − 0.866·12-s − 2.21·13-s − 3.09·15-s − 1/4·16-s − 0.485·17-s + 1.41·18-s − 0.458·19-s + 0.894·20-s − 0.852·22-s − 2.08·23-s − 0.612·24-s + 3/5·25-s − 1.56·26-s + 1.92·27-s − 1.11·29-s − 2.19·30-s + 0.359·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\,\Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(3^{3} \cdot 7^{6} \cdot 41^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6027} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(6,\ 3^{3} \cdot 7^{6} \cdot 41^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;41\}$, \(F_p(T)\) is a polynomial of degree 6. If $p \in \{3,\;7,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 - T )^{3} \)
7 \( 1 \)
41$C_1$ \( ( 1 + T )^{3} \)
good2$S_4\times C_2$ \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
5$S_4\times C_2$ \( 1 + 4 T + 13 T^{2} + 36 T^{3} + 13 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 4 T + 34 T^{2} + 84 T^{3} + 34 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 8 T + 53 T^{2} + 204 T^{3} + 53 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 2 T + 28 T^{2} + 6 T^{3} + 28 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 2 T + 51 T^{2} + 68 T^{3} + 51 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 10 T + 95 T^{2} + 476 T^{3} + 95 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 6 T + 60 T^{2} + 262 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 2 T + 2 T^{2} + 132 T^{3} + 2 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 20 T + 228 T^{2} - 1646 T^{3} + 228 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 10 T + 10 T^{2} + 296 T^{3} + 10 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 4 T + 106 T^{2} + 384 T^{3} + 106 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 14 T + 3 p T^{2} - 1452 T^{3} + 3 p^{2} T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 8 T + 137 T^{2} - 976 T^{3} + 137 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 8 T + 188 T^{2} - 930 T^{3} + 188 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 12 T + 77 T^{2} - 632 T^{3} + 77 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 32 T + 550 T^{2} + 5712 T^{3} + 550 p T^{4} + 32 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 4 T + 120 T^{2} + 130 T^{3} + 120 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 20 T + 305 T^{2} + 3192 T^{3} + 305 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 14 T + 259 T^{2} - 2028 T^{3} + 259 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 14 T + 263 T^{2} + 2308 T^{3} + 263 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 12 T + 305 T^{2} - 2180 T^{3} + 305 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.51584899694020341119861063282, −7.42316092050796779346889251448, −7.24513233272592872275306932976, −7.03435185340450888653458762601, −6.62785510414224473596620669061, −6.27911594932678334296821837468, −6.03705165640759462958593198826, −5.76165040722357987498405445808, −5.44925842869473204003440256971, −5.32721283559812962615037161175, −4.71941309232517034831688769665, −4.62849428651522142898956675643, −4.55456438716255186101379185128, −4.26276476813171114000745602022, −3.99814127938129766150250422065, −3.91994287805849195431874347760, −3.50317544407652326755331625285, −3.49238255870543996453902242053, −2.74430060367268899706109205095, −2.68790432287775649397997189871, −2.34783463452892786753153583100, −2.27315507031838323711502170154, −2.16916735838813378089131663799, −1.25011270715050356204940204496, −1.07130706863150774851147310589, 0, 0, 0, 1.07130706863150774851147310589, 1.25011270715050356204940204496, 2.16916735838813378089131663799, 2.27315507031838323711502170154, 2.34783463452892786753153583100, 2.68790432287775649397997189871, 2.74430060367268899706109205095, 3.49238255870543996453902242053, 3.50317544407652326755331625285, 3.91994287805849195431874347760, 3.99814127938129766150250422065, 4.26276476813171114000745602022, 4.55456438716255186101379185128, 4.62849428651522142898956675643, 4.71941309232517034831688769665, 5.32721283559812962615037161175, 5.44925842869473204003440256971, 5.76165040722357987498405445808, 6.03705165640759462958593198826, 6.27911594932678334296821837468, 6.62785510414224473596620669061, 7.03435185340450888653458762601, 7.24513233272592872275306932976, 7.42316092050796779346889251448, 7.51584899694020341119861063282

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