Properties

Label 6-140e3-1.1-c5e3-0-1
Degree $6$
Conductor $2744000$
Sign $1$
Analytic cond. $11320.5$
Root an. cond. $4.73853$
Motivic weight $5$
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  + 6·3-s + 75·5-s + 147·7-s − 206·9-s − 14·11-s − 4·13-s + 450·15-s + 44·17-s + 2.32e3·19-s + 882·21-s + 3.67e3·23-s + 3.75e3·25-s − 552·27-s + 4.09e3·29-s + 5.88e3·31-s − 84·33-s + 1.10e4·35-s + 1.13e4·37-s − 24·39-s + 1.14e4·41-s + 1.85e4·43-s − 1.54e4·45-s + 2.17e4·47-s + 1.44e4·49-s + 264·51-s + 7.49e3·53-s − 1.05e3·55-s + ⋯
L(s)  = 1  + 0.384·3-s + 1.34·5-s + 1.13·7-s − 0.847·9-s − 0.0348·11-s − 0.00656·13-s + 0.516·15-s + 0.0369·17-s + 1.47·19-s + 0.436·21-s + 1.44·23-s + 6/5·25-s − 0.145·27-s + 0.903·29-s + 1.10·31-s − 0.0134·33-s + 1.52·35-s + 1.36·37-s − 0.00252·39-s + 1.06·41-s + 1.52·43-s − 1.13·45-s + 1.43·47-s + 6/7·49-s + 0.0142·51-s + 0.366·53-s − 0.0468·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2744000 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2744000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2744000\)    =    \(2^{6} \cdot 5^{3} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(11320.5\)
Root analytic conductor: \(4.73853\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2744000,\ (\ :5/2, 5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(9.472754645\)
\(L(\frac12)\) \(\approx\) \(9.472754645\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 - p^{2} T )^{3} \)
7$C_1$ \( ( 1 - p^{2} T )^{3} \)
good3$S_4\times C_2$ \( 1 - 2 p T + 242 T^{2} - 712 p T^{3} + 242 p^{5} T^{4} - 2 p^{11} T^{5} + p^{15} T^{6} \)
11$S_4\times C_2$ \( 1 + 14 T + 335922 T^{2} + 16946168 T^{3} + 335922 p^{5} T^{4} + 14 p^{10} T^{5} + p^{15} T^{6} \)
13$S_4\times C_2$ \( 1 + 4 T + 688508 T^{2} - 3170390 T^{3} + 688508 p^{5} T^{4} + 4 p^{10} T^{5} + p^{15} T^{6} \)
17$S_4\times C_2$ \( 1 - 44 T + 11856 p^{2} T^{2} + 154171654 T^{3} + 11856 p^{7} T^{4} - 44 p^{10} T^{5} + p^{15} T^{6} \)
19$S_4\times C_2$ \( 1 - 2328 T + 2524317 T^{2} - 99195680 T^{3} + 2524317 p^{5} T^{4} - 2328 p^{10} T^{5} + p^{15} T^{6} \)
23$S_4\times C_2$ \( 1 - 3676 T + 21311145 T^{2} - 45236333032 T^{3} + 21311145 p^{5} T^{4} - 3676 p^{10} T^{5} + p^{15} T^{6} \)
29$S_4\times C_2$ \( 1 - 4092 T + 44125116 T^{2} - 150282374694 T^{3} + 44125116 p^{5} T^{4} - 4092 p^{10} T^{5} + p^{15} T^{6} \)
31$S_4\times C_2$ \( 1 - 5888 T + 1430819 p T^{2} - 125334777344 T^{3} + 1430819 p^{6} T^{4} - 5888 p^{10} T^{5} + p^{15} T^{6} \)
37$S_4\times C_2$ \( 1 - 11378 T + 249081611 T^{2} - 1625278009292 T^{3} + 249081611 p^{5} T^{4} - 11378 p^{10} T^{5} + p^{15} T^{6} \)
41$S_4\times C_2$ \( 1 - 11450 T + 303190107 T^{2} - 2174527049300 T^{3} + 303190107 p^{5} T^{4} - 11450 p^{10} T^{5} + p^{15} T^{6} \)
43$S_4\times C_2$ \( 1 - 18544 T + 427949605 T^{2} - 4701687457808 T^{3} + 427949605 p^{5} T^{4} - 18544 p^{10} T^{5} + p^{15} T^{6} \)
47$S_4\times C_2$ \( 1 - 21754 T + 817528902 T^{2} - 10151999940700 T^{3} + 817528902 p^{5} T^{4} - 21754 p^{10} T^{5} + p^{15} T^{6} \)
53$S_4\times C_2$ \( 1 - 7494 T + 1092946479 T^{2} - 5524002791484 T^{3} + 1092946479 p^{5} T^{4} - 7494 p^{10} T^{5} + p^{15} T^{6} \)
59$S_4\times C_2$ \( 1 - 12388 T + 210415569 T^{2} + 23463075596456 T^{3} + 210415569 p^{5} T^{4} - 12388 p^{10} T^{5} + p^{15} T^{6} \)
61$S_4\times C_2$ \( 1 - 27182 T + 2359167815 T^{2} - 46085588664044 T^{3} + 2359167815 p^{5} T^{4} - 27182 p^{10} T^{5} + p^{15} T^{6} \)
67$S_4\times C_2$ \( 1 + 7676 T + 23794675 p T^{2} + 5556434935144 T^{3} + 23794675 p^{6} T^{4} + 7676 p^{10} T^{5} + p^{15} T^{6} \)
71$S_4\times C_2$ \( 1 - 81992 T + 4847939253 T^{2} - 182439241075184 T^{3} + 4847939253 p^{5} T^{4} - 81992 p^{10} T^{5} + p^{15} T^{6} \)
73$S_4\times C_2$ \( 1 - 230 T - 689976937 T^{2} - 11997243597140 T^{3} - 689976937 p^{5} T^{4} - 230 p^{10} T^{5} + p^{15} T^{6} \)
79$S_4\times C_2$ \( 1 + 15926 T + 766889110 T^{2} - 176082125424572 T^{3} + 766889110 p^{5} T^{4} + 15926 p^{10} T^{5} + p^{15} T^{6} \)
83$S_4\times C_2$ \( 1 + 86100 T + 12298902585 T^{2} + 638013375785400 T^{3} + 12298902585 p^{5} T^{4} + 86100 p^{10} T^{5} + p^{15} T^{6} \)
89$S_4\times C_2$ \( 1 + 95710 T + 14224701483 T^{2} + 763443390521980 T^{3} + 14224701483 p^{5} T^{4} + 95710 p^{10} T^{5} + p^{15} T^{6} \)
97$S_4\times C_2$ \( 1 + 176188 T + 27195006632 T^{2} + 2984320016065750 T^{3} + 27195006632 p^{5} T^{4} + 176188 p^{10} T^{5} + p^{15} T^{6} \)
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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

−11.15155187515066393564973953431, −10.41707015427650963389091043908, −10.26603845214062733686387968432, −9.757465862979467932520795011601, −9.347009442680675662037073709182, −9.180044296371370195865357031808, −8.885996551316146855696088528458, −8.311407323172569179001205108894, −8.054727619267592753293839216260, −7.78948134162268314520418813995, −7.22511552429934842807447086281, −6.67397800399970079603551464943, −6.63160997767520776506860706389, −5.73927544067276422839766454541, −5.52203918705780333620702924555, −5.40568048479750788843420185290, −4.82714036596842240427396885373, −4.23520268526526143041820175965, −3.92346835388940290120785466087, −2.74191624921347002391444778744, −2.70761496771098044593245423352, −2.58914842286890635991602355836, −1.47287247417362097021632997009, −1.08545209538125062075450240043, −0.72920375490633428883223405250, 0.72920375490633428883223405250, 1.08545209538125062075450240043, 1.47287247417362097021632997009, 2.58914842286890635991602355836, 2.70761496771098044593245423352, 2.74191624921347002391444778744, 3.92346835388940290120785466087, 4.23520268526526143041820175965, 4.82714036596842240427396885373, 5.40568048479750788843420185290, 5.52203918705780333620702924555, 5.73927544067276422839766454541, 6.63160997767520776506860706389, 6.67397800399970079603551464943, 7.22511552429934842807447086281, 7.78948134162268314520418813995, 8.054727619267592753293839216260, 8.311407323172569179001205108894, 8.885996551316146855696088528458, 9.180044296371370195865357031808, 9.347009442680675662037073709182, 9.757465862979467932520795011601, 10.26603845214062733686387968432, 10.41707015427650963389091043908, 11.15155187515066393564973953431

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