Properties

Degree $6$
Conductor $832706445312$
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

Degree: \(6\)
Conductor: \(2^{18} \cdot 3^{3} \cdot 7^{6}\)
Sign: $-1$
Motivic weight: \(1\)
Character: induced by $\chi_{9408} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{18} \cdot 3^{3} \cdot 7^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.34020726946861713556109191396, −7.20405945165639230519978538402, −6.75587048098838539261431354111, −6.40801437943889114175738642018, −6.20206591827279172002310625883, −6.17267145425137821830976439875, −5.84180859648746657195865322276, −5.44671339569784654178691637401, −5.23054137436340224665059310295, −5.11233823799644057809140436418, −4.62870180867377877754050293402, −4.46217330808684756169947671372, −4.40289111494656055123202129785, −3.91007278783536783889195190449, −3.78719663386081581360766789617, −3.67229119775004811894662687036, −3.16635104865747422298085471866, −2.99588917665941883418342124759, −2.90278470694470282569081857034, −2.50756424549314561971395587214, −2.02668273010612640987199235782, −1.91715581037774558905358615271, −1.76907924265311502645732162474, −1.52775281735858768905543728517, −1.09315840968816290695441560999, 0, 0, 0, 1.09315840968816290695441560999, 1.52775281735858768905543728517, 1.76907924265311502645732162474, 1.91715581037774558905358615271, 2.02668273010612640987199235782, 2.50756424549314561971395587214, 2.90278470694470282569081857034, 2.99588917665941883418342124759, 3.16635104865747422298085471866, 3.67229119775004811894662687036, 3.78719663386081581360766789617, 3.91007278783536783889195190449, 4.40289111494656055123202129785, 4.46217330808684756169947671372, 4.62870180867377877754050293402, 5.11233823799644057809140436418, 5.23054137436340224665059310295, 5.44671339569784654178691637401, 5.84180859648746657195865322276, 6.17267145425137821830976439875, 6.20206591827279172002310625883, 6.40801437943889114175738642018, 6.75587048098838539261431354111, 7.20405945165639230519978538402, 7.34020726946861713556109191396

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