Properties

Degree 6
Conductor $ 2^{18} \cdot 3^{3} \cdot 7^{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  − 3·3-s + 6·9-s + 3·13-s + 6·17-s − 3·19-s − 6·23-s − 6·25-s − 10·27-s − 12·29-s + 3·31-s − 3·37-s − 9·39-s − 6·41-s − 15·43-s + 12·47-s − 18·51-s − 6·53-s + 9·57-s − 12·59-s + 18·61-s − 9·67-s + 18·69-s + 33·73-s + 18·75-s − 27·79-s + 15·81-s − 18·83-s + ⋯
L(s)  = 1  − 1.73·3-s + 2·9-s + 0.832·13-s + 1.45·17-s − 0.688·19-s − 1.25·23-s − 6/5·25-s − 1.92·27-s − 2.22·29-s + 0.538·31-s − 0.493·37-s − 1.44·39-s − 0.937·41-s − 2.28·43-s + 1.75·47-s − 2.52·51-s − 0.824·53-s + 1.19·57-s − 1.56·59-s + 2.30·61-s − 1.09·67-s + 2.16·69-s + 3.86·73-s + 2.07·75-s − 3.03·79-s + 5/3·81-s − 1.97·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(2^{18} \cdot 3^{3} \cdot 7^{6}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{9408} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(3\)
Selberg data  =  \((6,\ 2^{18} \cdot 3^{3} \cdot 7^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 6. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{3} \)
7 \( 1 \)
good5$S_4\times C_2$ \( 1 + 6 T^{2} - 4 T^{3} + 6 p T^{4} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 6 T^{2} - 38 T^{3} + 6 p T^{4} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 3 T + 3 T^{2} + 34 T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 6 T + 27 T^{2} - 108 T^{3} + 27 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 3 T + 21 T^{2} + 2 T^{3} + 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 6 T + 45 T^{2} + 180 T^{3} + 45 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 12 T + 126 T^{2} + 728 T^{3} + 126 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 3 T + 72 T^{2} - 139 T^{3} + 72 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 3 T + 27 T^{2} - 146 T^{3} + 27 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 6 T + 27 T^{2} - 20 T^{3} + 27 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 15 T + 177 T^{2} + 1318 T^{3} + 177 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 12 T + 153 T^{2} - 1016 T^{3} + 153 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 6 T + 78 T^{2} + 802 T^{3} + 78 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 12 T + 198 T^{2} + 1334 T^{3} + 198 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 18 T + 195 T^{2} - 1644 T^{3} + 195 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 9 T + 189 T^{2} + 1042 T^{3} + 189 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 177 T^{2} - 32 T^{3} + 177 p T^{4} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 33 T + 555 T^{2} - 5814 T^{3} + 555 p T^{4} - 33 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 27 T + 432 T^{2} + 4619 T^{3} + 432 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 18 T + 234 T^{2} + 2040 T^{3} + 234 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 12 T + 279 T^{2} + 2024 T^{3} + 279 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 264 T^{2} + 38 T^{3} + 264 p T^{4} + p^{3} T^{6} \)
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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

−7.16994662981887554743564024795, −6.92409048234475903406407757597, −6.51021596586698558151751970160, −6.49729110467948366785857685470, −6.07542877888881999609531208228, −6.00964920184741095216465889215, −5.78946789290990515161331899952, −5.52801368545886163148801705361, −5.35521093745068329349696699794, −5.29618087820360876087910651744, −4.84321290027232170982453048277, −4.68055038376497631588766297219, −4.33243595973469640679492699765, −3.93312934682435548703702169303, −3.92472690580912264738824293409, −3.79714993411567762703814108660, −3.34312123938844212659344293351, −3.21401740570025089886595575227, −2.80303143413862762195587299371, −2.21455461330791820123463672040, −2.14930174716000056105808740955, −1.80852741423213065060068507172, −1.33821685639351681782868154790, −1.32039026572221818321214680682, −0.927590678553471684916812242098, 0, 0, 0, 0.927590678553471684916812242098, 1.32039026572221818321214680682, 1.33821685639351681782868154790, 1.80852741423213065060068507172, 2.14930174716000056105808740955, 2.21455461330791820123463672040, 2.80303143413862762195587299371, 3.21401740570025089886595575227, 3.34312123938844212659344293351, 3.79714993411567762703814108660, 3.92472690580912264738824293409, 3.93312934682435548703702169303, 4.33243595973469640679492699765, 4.68055038376497631588766297219, 4.84321290027232170982453048277, 5.29618087820360876087910651744, 5.35521093745068329349696699794, 5.52801368545886163148801705361, 5.78946789290990515161331899952, 6.00964920184741095216465889215, 6.07542877888881999609531208228, 6.49729110467948366785857685470, 6.51021596586698558151751970160, 6.92409048234475903406407757597, 7.16994662981887554743564024795

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