Properties

Label 6-8400e3-1.1-c1e3-0-2
Degree $6$
Conductor $592704000000$
Sign $1$
Analytic cond. $301765.$
Root an. cond. $8.18989$
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  + 3·3-s + 3·7-s + 6·9-s − 2·11-s − 2·13-s − 2·19-s + 9·21-s + 8·23-s + 10·27-s + 2·29-s + 2·31-s − 6·33-s + 4·37-s − 6·39-s + 14·41-s + 12·43-s + 8·47-s + 6·49-s − 14·53-s − 6·57-s − 8·59-s + 2·61-s + 18·63-s + 24·69-s + 6·71-s − 6·73-s − 6·77-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.13·7-s + 2·9-s − 0.603·11-s − 0.554·13-s − 0.458·19-s + 1.96·21-s + 1.66·23-s + 1.92·27-s + 0.371·29-s + 0.359·31-s − 1.04·33-s + 0.657·37-s − 0.960·39-s + 2.18·41-s + 1.82·43-s + 1.16·47-s + 6/7·49-s − 1.92·53-s − 0.794·57-s − 1.04·59-s + 0.256·61-s + 2.26·63-s + 2.88·69-s + 0.712·71-s − 0.702·73-s − 0.683·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(301765.\)
Root analytic conductor: \(8.18989\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(18.34151191\)
\(L(\frac12)\) \(\approx\) \(18.34151191\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
5 \( 1 \)
7$C_1$ \( ( 1 - T )^{3} \)
good11$S_4\times C_2$ \( 1 + 2 T + 13 T^{2} + 36 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
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$S_4\times C_2$ \( 1 + 35 T^{2} - 16 T^{3} + 35 p T^{4} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 2 T + 45 T^{2} + 68 T^{3} + 45 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 8 T + 77 T^{2} - 352 T^{3} + 77 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
29$D_{6}$ \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 2 T + 81 T^{2} - 116 T^{3} + 81 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
37$D_{6}$ \( 1 - 4 T + 63 T^{2} - 232 T^{3} + 63 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 14 T + 175 T^{2} - 1188 T^{3} + 175 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 12 T + 113 T^{2} - 712 T^{3} + 113 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 8 T + 77 T^{2} - 496 T^{3} + 77 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 14 T + 131 T^{2} + 1012 T^{3} + 131 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 8 T + 113 T^{2} + 688 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 2 T + 35 T^{2} - 780 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
67$C_2$ \( ( 1 + p T^{2} )^{3} \)
71$S_4\times C_2$ \( 1 - 6 T + 113 T^{2} - 1052 T^{3} + 113 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 6 T + 95 T^{2} + 116 T^{3} + 95 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 20 T + 317 T^{2} - 3096 T^{3} + 317 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 8 T + 185 T^{2} - 1072 T^{3} + 185 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 2 T + 207 T^{2} - 156 T^{3} + 207 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 14 T + 295 T^{2} - 2372 T^{3} + 295 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.19444929522728633540087632574, −6.49066396373267745474052529569, −6.47364368908625904559807727802, −6.45423508559015539058118507184, −5.82828512353959628894767772970, −5.70998391123050498846119831687, −5.61948946757189128792670040633, −4.98314834828739987860795430481, −4.86336668371327873789390377295, −4.78416831151292123118871160746, −4.44838441018111289485515462988, −4.33507909732403557736725737123, −4.12209253309611255921931639630, −3.56074499534995349318177843388, −3.37139725257143495796766936367, −3.33877670718866482812961149484, −2.79295555995698333158212282310, −2.60290668251388933044722838760, −2.48236491776594138227383746763, −2.13322406136382921097060254148, −1.85458639150356205834873124667, −1.67182960224454113879544873335, −0.949326910976761379206311139407, −0.822791417782459046296341571423, −0.62685474069221349741778766067, 0.62685474069221349741778766067, 0.822791417782459046296341571423, 0.949326910976761379206311139407, 1.67182960224454113879544873335, 1.85458639150356205834873124667, 2.13322406136382921097060254148, 2.48236491776594138227383746763, 2.60290668251388933044722838760, 2.79295555995698333158212282310, 3.33877670718866482812961149484, 3.37139725257143495796766936367, 3.56074499534995349318177843388, 4.12209253309611255921931639630, 4.33507909732403557736725737123, 4.44838441018111289485515462988, 4.78416831151292123118871160746, 4.86336668371327873789390377295, 4.98314834828739987860795430481, 5.61948946757189128792670040633, 5.70998391123050498846119831687, 5.82828512353959628894767772970, 6.45423508559015539058118507184, 6.47364368908625904559807727802, 6.49066396373267745474052529569, 7.19444929522728633540087632574

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