Properties

Degree 6
Conductor $ 3^{6} \cdot 7^{3} \cdot 11^{6} $
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 − 4-s − 5-s − 3·7-s − 8-s − 10-s − 7·13-s − 3·14-s − 16-s + 17-s − 8·19-s + 20-s + 2·23-s − 7·25-s − 7·26-s + 3·28-s + 7·29-s − 2·31-s − 32-s + 34-s + 3·35-s − 3·37-s − 8·38-s + 40-s + 13·41-s − 8·43-s + 2·46-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.13·7-s − 0.353·8-s − 0.316·10-s − 1.94·13-s − 0.801·14-s − 1/4·16-s + 0.242·17-s − 1.83·19-s + 0.223·20-s + 0.417·23-s − 7/5·25-s − 1.37·26-s + 0.566·28-s + 1.29·29-s − 0.359·31-s − 0.176·32-s + 0.171·34-s + 0.507·35-s − 0.493·37-s − 1.29·38-s + 0.158·40-s + 2.03·41-s − 1.21·43-s + 0.294·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(3^{6} \cdot 7^{3} \cdot 11^{6}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{7623} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(6,\ 3^{6} \cdot 7^{3} \cdot 11^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\(F_p(T)\) is a polynomial of degree 6. If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
7$C_1$ \( ( 1 + T )^{3} \)
11 \( 1 \)
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$S_4\times C_2$ \( 1 + T + 8 T^{2} + 11 T^{3} + 8 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 7 T + 51 T^{2} + 184 T^{3} + 51 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - T + 44 T^{2} - 35 T^{3} + 44 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 8 T + 63 T^{2} + 260 T^{3} + 63 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 2 T - 9 T^{2} + 60 T^{3} - 9 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 7 T + 99 T^{2} - 408 T^{3} + 99 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 2 T + 65 T^{2} + 132 T^{3} + 65 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 3 T + 107 T^{2} + 214 T^{3} + 107 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 13 T + 163 T^{2} - 1098 T^{3} + 163 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 8 T + 58 T^{2} + 266 T^{3} + 58 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 6 T + 110 T^{2} - 530 T^{3} + 110 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 21 T + 299 T^{2} - 2522 T^{3} + 299 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 6 T + 150 T^{2} + 622 T^{3} + 150 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 12 T + 165 T^{2} + 1172 T^{3} + 165 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 2 T + 50 T^{2} - 852 T^{3} + 50 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 179 T^{2} + 76 T^{3} + 179 p T^{4} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 4 T - 7 T^{2} + 1044 T^{3} - 7 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 18 T + 209 T^{2} + 1636 T^{3} + 209 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 12 T + 78 T^{2} - 64 T^{3} + 78 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - T + 188 T^{2} - 251 T^{3} + 188 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 25 T + 315 T^{2} + 2968 T^{3} + 315 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.32243093458217316445378590205, −7.02018855868030406447974577884, −6.87467957467699686101175501261, −6.63081503260843206897252596232, −6.38315974997757856857518105402, −6.08730902441472551775736122778, −5.76843724492194050768891434964, −5.61632422707582429210251605808, −5.60751186879212159484689771214, −5.10661737654976694723737557982, −4.74493085311436246762915935773, −4.70286962422610905487013405166, −4.38149237771943922601639589606, −4.27802983598255640921736603340, −3.96865714821898277233893495545, −3.80481081887071681648686194081, −3.48287112928752498609953364640, −3.18184861399603950120041373680, −2.69123748340800522429160292847, −2.58977444438676861200116652286, −2.53206740589611133411818548199, −2.06619773444085077015391938485, −1.75885500882358783873392697265, −1.14906179842163953689167590250, −0.906686163267401990341140566352, 0, 0, 0, 0.906686163267401990341140566352, 1.14906179842163953689167590250, 1.75885500882358783873392697265, 2.06619773444085077015391938485, 2.53206740589611133411818548199, 2.58977444438676861200116652286, 2.69123748340800522429160292847, 3.18184861399603950120041373680, 3.48287112928752498609953364640, 3.80481081887071681648686194081, 3.96865714821898277233893495545, 4.27802983598255640921736603340, 4.38149237771943922601639589606, 4.70286962422610905487013405166, 4.74493085311436246762915935773, 5.10661737654976694723737557982, 5.60751186879212159484689771214, 5.61632422707582429210251605808, 5.76843724492194050768891434964, 6.08730902441472551775736122778, 6.38315974997757856857518105402, 6.63081503260843206897252596232, 6.87467957467699686101175501261, 7.02018855868030406447974577884, 7.32243093458217316445378590205

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