Properties

Label 6-9680e3-1.1-c1e3-0-5
Degree $6$
Conductor $907039232000$
Sign $1$
Analytic cond. $461803.$
Root an. cond. $8.79176$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·5-s − 7-s − 9-s + 2·13-s + 3·15-s + 4·17-s + 8·19-s + 21-s + 2·23-s + 6·25-s + 6·27-s + 14·29-s + 2·31-s + 3·35-s + 4·37-s − 2·39-s + 11·41-s − 29·43-s + 3·45-s − 47-s − 15·49-s − 4·51-s + 10·53-s − 8·57-s − 6·59-s − 11·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s − 0.377·7-s − 1/3·9-s + 0.554·13-s + 0.774·15-s + 0.970·17-s + 1.83·19-s + 0.218·21-s + 0.417·23-s + 6/5·25-s + 1.15·27-s + 2.59·29-s + 0.359·31-s + 0.507·35-s + 0.657·37-s − 0.320·39-s + 1.71·41-s − 4.42·43-s + 0.447·45-s − 0.145·47-s − 2.14·49-s − 0.560·51-s + 1.37·53-s − 1.05·57-s − 0.781·59-s − 1.40·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 5^{3} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(461803.\)
Root analytic conductor: \(8.79176\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{12} \cdot 5^{3} \cdot 11^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.026618995\)
\(L(\frac12)\) \(\approx\) \(3.026618995\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{3} \)
11 \( 1 \)
good3$S_4\times C_2$ \( 1 + T + 2 T^{2} - p T^{3} + 2 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + T + 16 T^{2} + 15 T^{3} + 16 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 2 T - T^{2} + 32 T^{3} - p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 4 T + 35 T^{2} - 88 T^{3} + 35 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 8 T + 37 T^{2} - 156 T^{3} + 37 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 2 T + T^{2} + 176 T^{3} + p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 14 T + 123 T^{2} - 724 T^{3} + 123 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 2 T + 77 T^{2} - 128 T^{3} + 77 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 4 T + 87 T^{2} - 312 T^{3} + 87 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 11 T + 146 T^{2} - 903 T^{3} + 146 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 29 T + 404 T^{2} + 3343 T^{3} + 404 p T^{4} + 29 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + T + 82 T^{2} + 17 T^{3} + 82 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 10 T + 135 T^{2} - 1064 T^{3} + 135 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 6 T + 117 T^{2} + 464 T^{3} + 117 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 11 T + 154 T^{2} + 15 p T^{3} + 154 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 7 T + 42 T^{2} + 551 T^{3} + 42 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 6 T + 217 T^{2} - 840 T^{3} + 217 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 4 T - 5 T^{2} + 984 T^{3} - 5 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 6 T + 73 T^{2} + 556 T^{3} + 73 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 6 T + 189 T^{2} - 752 T^{3} + 189 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 13 T + 302 T^{2} + 2281 T^{3} + 302 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 22 T + 411 T^{2} - 4416 T^{3} + 411 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
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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

−6.53928602821090878973529778583, −6.51031951813545036848551102635, −6.44389270812691826100822377372, −6.35601010890772905824442510880, −5.79938806561774021213635050176, −5.55862083312608906415745517244, −5.49483857237656444000423265018, −5.03352270953383442003103383735, −4.79935517826363527055032857700, −4.76910853612375288725464342238, −4.65939931831764024063808780571, −4.21504638443584018838846873101, −3.91278903982569370431709159834, −3.54747278973172317546332635735, −3.32913993957591957224032134527, −3.31061352988747627995593533045, −2.83456073816353608845409011958, −2.77041315277758713529678592268, −2.71046324923964394527678857508, −1.82737595508828089698580122775, −1.58115792541633857627551091148, −1.32370795982289362087531172755, −0.940247053199544369288214000763, −0.51642213846530661334004078393, −0.47152962199342511796424097722, 0.47152962199342511796424097722, 0.51642213846530661334004078393, 0.940247053199544369288214000763, 1.32370795982289362087531172755, 1.58115792541633857627551091148, 1.82737595508828089698580122775, 2.71046324923964394527678857508, 2.77041315277758713529678592268, 2.83456073816353608845409011958, 3.31061352988747627995593533045, 3.32913993957591957224032134527, 3.54747278973172317546332635735, 3.91278903982569370431709159834, 4.21504638443584018838846873101, 4.65939931831764024063808780571, 4.76910853612375288725464342238, 4.79935517826363527055032857700, 5.03352270953383442003103383735, 5.49483857237656444000423265018, 5.55862083312608906415745517244, 5.79938806561774021213635050176, 6.35601010890772905824442510880, 6.44389270812691826100822377372, 6.51031951813545036848551102635, 6.53928602821090878973529778583

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