Properties

Degree $6$
Conductor $832706445312$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

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: \(0\)
Selberg data: \((6,\ 2^{18} \cdot 3^{3} \cdot 7^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6924407583\)
\(L(\frac12)\) \(\approx\) \(0.6924407583\)
\(L(\frac{3}{2})\) 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

−6.83703970916781063670435219124, −6.27689635819207242095295202085, −6.24932285746250591686561085507, −6.19012410509135295958823454038, −5.85797746772003044989683628698, −5.81752200660794771218579528723, −5.54145039310558001693901479501, −5.02811541677113932625704160923, −5.00293808712220904094785332711, −4.72736072426378905115018571416, −4.47503840107020430221377010288, −4.34998824963240448053312510011, −4.17324403729266268787230267666, −3.72976749159973237416133366956, −3.43122749814822771684727054663, −3.41981924068508168317564260184, −2.71418655918622414216178887001, −2.52847211601382724989689382595, −2.34435264276584154807412118641, −2.00786063452202614008030617818, −1.55022770192469998420233780204, −1.51735016009346723398151952212, −0.900993837304344759186486518542, −0.48434694327171247333537163029, −0.24115719727153709893694463870, 0.24115719727153709893694463870, 0.48434694327171247333537163029, 0.900993837304344759186486518542, 1.51735016009346723398151952212, 1.55022770192469998420233780204, 2.00786063452202614008030617818, 2.34435264276584154807412118641, 2.52847211601382724989689382595, 2.71418655918622414216178887001, 3.41981924068508168317564260184, 3.43122749814822771684727054663, 3.72976749159973237416133366956, 4.17324403729266268787230267666, 4.34998824963240448053312510011, 4.47503840107020430221377010288, 4.72736072426378905115018571416, 5.00293808712220904094785332711, 5.02811541677113932625704160923, 5.54145039310558001693901479501, 5.81752200660794771218579528723, 5.85797746772003044989683628698, 6.19012410509135295958823454038, 6.24932285746250591686561085507, 6.27689635819207242095295202085, 6.83703970916781063670435219124

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