Properties

Degree 6
Conductor $ 2^{9} \cdot 7^{3} \cdot 11^{3} \cdot 13^{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  + 2·3-s + 6·5-s − 3·7-s − 9-s − 3·11-s + 3·13-s + 12·15-s − 13·17-s + 19-s − 6·21-s − 13·23-s + 13·25-s − 7·27-s + 8·29-s − 17·31-s − 6·33-s − 18·35-s + 7·37-s + 6·39-s − 10·41-s + 9·43-s − 6·45-s − 2·47-s + 6·49-s − 26·51-s − 11·53-s − 18·55-s + ⋯
L(s)  = 1  + 1.15·3-s + 2.68·5-s − 1.13·7-s − 1/3·9-s − 0.904·11-s + 0.832·13-s + 3.09·15-s − 3.15·17-s + 0.229·19-s − 1.30·21-s − 2.71·23-s + 13/5·25-s − 1.34·27-s + 1.48·29-s − 3.05·31-s − 1.04·33-s − 3.04·35-s + 1.15·37-s + 0.960·39-s − 1.56·41-s + 1.37·43-s − 0.894·45-s − 0.291·47-s + 6/7·49-s − 3.64·51-s − 1.51·53-s − 2.42·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{9} \cdot 7^{3} \cdot 11^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{9} \cdot 7^{3} \cdot 11^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(2^{9} \cdot 7^{3} \cdot 11^{3} \cdot 13^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8008} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(6,\ 2^{9} \cdot 7^{3} \cdot 11^{3} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11,\;13\}$, \(F_p\) is a polynomial of degree 6. If $p \in \{2,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
7$C_1$ \( ( 1 + T )^{3} \)
11$C_1$ \( ( 1 + T )^{3} \)
13$C_1$ \( ( 1 - T )^{3} \)
good3$S_4\times C_2$ \( 1 - 2 T + 5 T^{2} - 5 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
5$S_4\times C_2$ \( 1 - 6 T + 23 T^{2} - 59 T^{3} + 23 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 13 T + 88 T^{2} + 414 T^{3} + 88 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - T + 42 T^{2} - 10 T^{3} + 42 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 13 T + 75 T^{2} + 330 T^{3} + 75 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 8 T + 59 T^{2} - 256 T^{3} + 59 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 17 T + 183 T^{2} + 1202 T^{3} + 183 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 7 T + 121 T^{2} - 514 T^{3} + 121 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 10 T - T^{2} - 364 T^{3} - p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 9 T + 120 T^{2} - 720 T^{3} + 120 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 2 T + 81 T^{2} - 36 T^{3} + 81 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 11 T + 98 T^{2} + 712 T^{3} + 98 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 9 T + 147 T^{2} - 866 T^{3} + 147 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 9 T + 206 T^{2} - 1114 T^{3} + 206 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 10 T + 219 T^{2} + 1303 T^{3} + 219 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 22 T + 5 p T^{2} + 3351 T^{3} + 5 p^{2} T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 18 T + 263 T^{2} + 2524 T^{3} + 263 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 3 T + 2 p T^{2} + 286 T^{3} + 2 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + T + 168 T^{2} - 108 T^{3} + 168 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 16 T + 295 T^{2} - 2607 T^{3} + 295 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - T + 35 T^{2} - 630 T^{3} + 35 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.29932741492675772219353689073, −7.02010479557464569627584213668, −6.64415420542117711965058257913, −6.50955220488829500330168221975, −6.20159089444985776303165611494, −6.06734875968744111386927469730, −6.02170469181912200130719556420, −5.71328393309283639644901701633, −5.53078847487313590767644337373, −5.36866081337490691864172793580, −4.95101429408696187818060667979, −4.57558867919856544924769673726, −4.44990077703980957017741540924, −4.01280813357970430282356885122, −3.88991272326361457793272365122, −3.69750902532459599530340402237, −3.12192397708819624298116069125, −3.02755822789317687495521529885, −2.74055194943842426213330460557, −2.32901207350468195015805086736, −2.29796990338717869773494490001, −2.14781565556594649594445845737, −1.71979115164282027571285931392, −1.59485021073848719015654026179, −1.14821143967501402151942184797, 0, 0, 0, 1.14821143967501402151942184797, 1.59485021073848719015654026179, 1.71979115164282027571285931392, 2.14781565556594649594445845737, 2.29796990338717869773494490001, 2.32901207350468195015805086736, 2.74055194943842426213330460557, 3.02755822789317687495521529885, 3.12192397708819624298116069125, 3.69750902532459599530340402237, 3.88991272326361457793272365122, 4.01280813357970430282356885122, 4.44990077703980957017741540924, 4.57558867919856544924769673726, 4.95101429408696187818060667979, 5.36866081337490691864172793580, 5.53078847487313590767644337373, 5.71328393309283639644901701633, 6.02170469181912200130719556420, 6.06734875968744111386927469730, 6.20159089444985776303165611494, 6.50955220488829500330168221975, 6.64415420542117711965058257913, 7.02010479557464569627584213668, 7.29932741492675772219353689073

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