Properties

Label 6-1080e3-1.1-c3e3-0-2
Degree $6$
Conductor $1259712000$
Sign $1$
Analytic cond. $258743.$
Root an. cond. $7.98261$
Motivic weight $3$
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  − 15·5-s − 24·7-s − 6·11-s + 48·13-s + 27·17-s − 195·19-s + 27·23-s + 150·25-s − 60·29-s − 279·31-s + 360·35-s − 138·37-s − 66·41-s + 222·43-s + 264·47-s − 345·49-s + 507·53-s + 90·55-s + 960·59-s + 543·61-s − 720·65-s − 1.08e3·67-s + 1.81e3·71-s + 1.36e3·73-s + 144·77-s − 129·79-s + 1.56e3·83-s + ⋯
L(s)  = 1  − 1.34·5-s − 1.29·7-s − 0.164·11-s + 1.02·13-s + 0.385·17-s − 2.35·19-s + 0.244·23-s + 6/5·25-s − 0.384·29-s − 1.61·31-s + 1.73·35-s − 0.613·37-s − 0.251·41-s + 0.787·43-s + 0.819·47-s − 1.00·49-s + 1.31·53-s + 0.220·55-s + 2.11·59-s + 1.13·61-s − 1.37·65-s − 1.98·67-s + 3.03·71-s + 2.18·73-s + 0.213·77-s − 0.183·79-s + 2.07·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{9} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(258743.\)
Root analytic conductor: \(7.98261\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 3^{9} \cdot 5^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(2.205666129\)
\(L(\frac12)\) \(\approx\) \(2.205666129\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + p T )^{3} \)
good7$S_4\times C_2$ \( 1 + 24 T + 921 T^{2} + 12904 T^{3} + 921 p^{3} T^{4} + 24 p^{6} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 + 6 T + 513 T^{2} + 60620 T^{3} + 513 p^{3} T^{4} + 6 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 - 48 T + 3435 T^{2} - 202552 T^{3} + 3435 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 - 27 T - 1158 T^{2} + 163141 T^{3} - 1158 p^{3} T^{4} - 27 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 + 195 T + 20832 T^{2} + 1720919 T^{3} + 20832 p^{3} T^{4} + 195 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 - 27 T - 168 T^{2} + 1130141 T^{3} - 168 p^{3} T^{4} - 27 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 + 60 T + 68067 T^{2} + 2620544 T^{3} + 68067 p^{3} T^{4} + 60 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 + 9 p T + 111132 T^{2} + 17041763 T^{3} + 111132 p^{3} T^{4} + 9 p^{7} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 + 138 T + 133923 T^{2} + 14102268 T^{3} + 133923 p^{3} T^{4} + 138 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 + 66 T + 63627 T^{2} - 6726804 T^{3} + 63627 p^{3} T^{4} + 66 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 - 222 T + 109929 T^{2} - 9784924 T^{3} + 109929 p^{3} T^{4} - 222 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 - 264 T + 124605 T^{2} - 55592752 T^{3} + 124605 p^{3} T^{4} - 264 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 - 507 T + 524166 T^{2} - 154386795 T^{3} + 524166 p^{3} T^{4} - 507 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 - 960 T + 843525 T^{2} - 409376120 T^{3} + 843525 p^{3} T^{4} - 960 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 - 543 T + 239946 T^{2} - 10210291 T^{3} + 239946 p^{3} T^{4} - 543 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 + 1086 T + 604461 T^{2} + 230866972 T^{3} + 604461 p^{3} T^{4} + 1086 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 - 1818 T + 2146461 T^{2} - 1508229108 T^{3} + 2146461 p^{3} T^{4} - 1818 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 - 1362 T + 1530147 T^{2} - 1054519260 T^{3} + 1530147 p^{3} T^{4} - 1362 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 + 129 T + 1152804 T^{2} + 62398365 T^{3} + 1152804 p^{3} T^{4} + 129 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 - 1569 T + 2366268 T^{2} - 1835088857 T^{3} + 2366268 p^{3} T^{4} - 1569 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 - 1770 T + 2057019 T^{2} - 1828728060 T^{3} + 2057019 p^{3} T^{4} - 1770 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 + 336 T + 1258275 T^{2} + 1117216928 T^{3} + 1258275 p^{3} T^{4} + 336 p^{6} T^{5} + p^{9} T^{6} \)
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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

−8.483441629752292970314887904792, −8.014376652796992595241418234995, −7.897674400161856714997380198595, −7.73444377317747625784313189391, −7.09860469169820502842328413137, −6.96763906413348885763739451779, −6.84685857118062351138210845052, −6.29185566504135210752834278145, −6.28647252428840953667154065897, −5.98732030337459700748122718794, −5.37343807838933192166919503595, −5.13599238330691984730084021693, −5.09757564732186598933390165989, −4.21435411314212050572730509551, −4.07454627461162653936112310997, −4.06583029608018745390308098173, −3.48363725633450228573488404023, −3.22063804103471025239991673774, −3.21959099103092146186817768481, −2.24109205251728923152830859612, −2.16498507685419549847179997503, −1.85088107026802371524457547911, −0.890618167532417611202929288593, −0.54481606115123228595255219766, −0.42664915158308489977791598156, 0.42664915158308489977791598156, 0.54481606115123228595255219766, 0.890618167532417611202929288593, 1.85088107026802371524457547911, 2.16498507685419549847179997503, 2.24109205251728923152830859612, 3.21959099103092146186817768481, 3.22063804103471025239991673774, 3.48363725633450228573488404023, 4.06583029608018745390308098173, 4.07454627461162653936112310997, 4.21435411314212050572730509551, 5.09757564732186598933390165989, 5.13599238330691984730084021693, 5.37343807838933192166919503595, 5.98732030337459700748122718794, 6.28647252428840953667154065897, 6.29185566504135210752834278145, 6.84685857118062351138210845052, 6.96763906413348885763739451779, 7.09860469169820502842328413137, 7.73444377317747625784313189391, 7.897674400161856714997380198595, 8.014376652796992595241418234995, 8.483441629752292970314887904792

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