Properties

Degree $6$
Conductor $104088305664$
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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{15} \cdot 3^{3} \cdot 7^{6}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{4704} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{15} \cdot 3^{3} \cdot 7^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5096249095\)
\(L(\frac12)\) \(\approx\) \(0.5096249095\)
\(L(\frac{3}{2})\) 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

−7.44373406966630828220882677052, −6.93285234944250783552437804799, −6.71856945037785703742892218012, −6.63379109654164686656357606432, −6.24034470587065198364617845050, −6.23427234536028422489273452084, −6.04654237563235028813913832507, −5.39985644997592997943725746179, −5.36094210299007151842193966620, −5.07923496502479215161310049106, −5.01698033129557396983080680309, −4.71549370602128257604884547947, −4.51151821972817572579110992570, −4.02495296902937986065097869810, −3.89419253645657089691542175921, −3.55177055865354894789649041411, −3.03190305017692665203277328688, −2.95240405114502665272081899105, −2.80570348642467528429465891883, −1.94788014867139412214005367440, −1.83968513706467725641536221696, −1.59777238552281127882857139504, −1.04364925395331376166529219278, −0.73679117860294609485670216937, −0.18733929018873429335510484084, 0.18733929018873429335510484084, 0.73679117860294609485670216937, 1.04364925395331376166529219278, 1.59777238552281127882857139504, 1.83968513706467725641536221696, 1.94788014867139412214005367440, 2.80570348642467528429465891883, 2.95240405114502665272081899105, 3.03190305017692665203277328688, 3.55177055865354894789649041411, 3.89419253645657089691542175921, 4.02495296902937986065097869810, 4.51151821972817572579110992570, 4.71549370602128257604884547947, 5.01698033129557396983080680309, 5.07923496502479215161310049106, 5.36094210299007151842193966620, 5.39985644997592997943725746179, 6.04654237563235028813913832507, 6.23427234536028422489273452084, 6.24034470587065198364617845050, 6.63379109654164686656357606432, 6.71856945037785703742892218012, 6.93285234944250783552437804799, 7.44373406966630828220882677052

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