Properties

Label 6-9280e3-1.1-c1e3-0-5
Degree $6$
Conductor $799178752000$
Sign $1$
Analytic cond. $406888.$
Root an. cond. $8.60820$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·5-s + 4·7-s − 9-s − 2·11-s + 2·13-s + 6·15-s − 4·17-s + 10·19-s − 8·21-s + 16·23-s + 6·25-s + 4·27-s + 3·29-s − 14·31-s + 4·33-s − 12·35-s + 8·37-s − 4·39-s − 2·41-s − 2·43-s + 3·45-s + 14·47-s − 5·49-s + 8·51-s − 6·53-s + 6·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.34·5-s + 1.51·7-s − 1/3·9-s − 0.603·11-s + 0.554·13-s + 1.54·15-s − 0.970·17-s + 2.29·19-s − 1.74·21-s + 3.33·23-s + 6/5·25-s + 0.769·27-s + 0.557·29-s − 2.51·31-s + 0.696·33-s − 2.02·35-s + 1.31·37-s − 0.640·39-s − 0.312·41-s − 0.304·43-s + 0.447·45-s + 2.04·47-s − 5/7·49-s + 1.12·51-s − 0.824·53-s + 0.809·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{18} \cdot 5^{3} \cdot 29^{3}\)
Sign: $1$
Analytic conductor: \(406888.\)
Root analytic conductor: \(8.60820\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{18} \cdot 5^{3} \cdot 29^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.215750188\)
\(L(\frac12)\) \(\approx\) \(2.215750188\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
29$C_1$ \( ( 1 - T )^{3} \)
good3$S_4\times C_2$ \( 1 + 2 T + 5 T^{2} + 8 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - 4 T + 3 p T^{2} - 52 T^{3} + 3 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 2 T + 25 T^{2} + 48 T^{3} + 25 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
13$D_{6}$ \( 1 - 2 T + 27 T^{2} - 44 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + 11 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 10 T + 85 T^{2} - 400 T^{3} + 85 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 16 T + 145 T^{2} - 36 p T^{3} + 145 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 14 T + 153 T^{2} + 944 T^{3} + 153 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 8 T + 87 T^{2} - 500 T^{3} + 87 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 2 T + 39 T^{2} + 396 T^{3} + 39 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 2 T - 3 T^{2} + 176 T^{3} - 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 14 T + 201 T^{2} - 1392 T^{3} + 201 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 6 T + 155 T^{2} + 628 T^{3} + 155 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 8 T + 113 T^{2} + 1024 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 6 T + 75 T^{2} - 516 T^{3} + 75 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 28 T + 453 T^{2} + 4468 T^{3} + 453 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 28 T + 389 T^{2} - 3704 T^{3} + 389 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 16 T + 119 T^{2} + 636 T^{3} + 119 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 6 T + 149 T^{2} + 488 T^{3} + 149 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 12 T + 3 p T^{2} + 1844 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 10 T + 279 T^{2} + 1740 T^{3} + 279 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 8 T + 223 T^{2} - 1628 T^{3} + 223 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
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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

−6.90999334003552050503973933910, −6.45926556688891729312368073593, −6.40253681745758976368405962927, −6.17543822606705571168957001112, −5.67658971145908870455401976258, −5.54606692283542104916671743893, −5.47471492372915927988487682742, −5.05275989356602903306789380517, −4.99722490067988967049466930732, −4.94598658445795776316211612627, −4.45275113824320270498635321924, −4.37670830407645773687979590693, −4.05980441164129616465328405337, −3.72084713703610837747245231267, −3.42925222716854963684195562971, −3.12682226251190366217709478394, −2.92550077845375672852216866059, −2.71571850331521682898704755128, −2.58187760091029573743935237252, −1.71699189980453273525072295488, −1.64613165348937317984221217019, −1.39106696634955503259554986641, −0.968173706905548842808269452568, −0.46867263416920681363894126338, −0.44703386411563720434539663410, 0.44703386411563720434539663410, 0.46867263416920681363894126338, 0.968173706905548842808269452568, 1.39106696634955503259554986641, 1.64613165348937317984221217019, 1.71699189980453273525072295488, 2.58187760091029573743935237252, 2.71571850331521682898704755128, 2.92550077845375672852216866059, 3.12682226251190366217709478394, 3.42925222716854963684195562971, 3.72084713703610837747245231267, 4.05980441164129616465328405337, 4.37670830407645773687979590693, 4.45275113824320270498635321924, 4.94598658445795776316211612627, 4.99722490067988967049466930732, 5.05275989356602903306789380517, 5.47471492372915927988487682742, 5.54606692283542104916671743893, 5.67658971145908870455401976258, 6.17543822606705571168957001112, 6.40253681745758976368405962927, 6.45926556688891729312368073593, 6.90999334003552050503973933910

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