Properties

Label 6-2160e3-1.1-c3e3-0-6
Degree $6$
Conductor $10077696000$
Sign $1$
Analytic cond. $2.06994\times 10^{6}$
Root an. cond. $11.2891$
Motivic weight $3$
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  − 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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 3^{9} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(2.06994\times 10^{6}\)
Root analytic conductor: \(11.2891\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{12} \cdot 3^{9} \cdot 5^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(7.056226172\)
\(L(\frac12)\) \(\approx\) \(7.056226172\)
\(L(\frac{5}{2})\) 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

−7.930585679341966779541506241138, −7.33926244443408543604324137248, −7.28494703873748847247197724141, −7.15579270891055384985061994738, −6.55141868324495107073112664495, −6.44646541365947202295326883839, −6.22872659077397790481094885941, −5.67890031403640141161490368175, −5.47193323223030545607493864514, −5.32227055090685237707065931162, −4.82477950212275281403964318734, −4.73111774103424182475517690557, −4.52433254976551109215919417086, −4.11312521269399621741835512475, −3.72519172146976864084251376904, −3.49416434873689011936539627471, −3.20721752477552417217794960705, −3.07978084709838559390847820032, −2.64414493181487465675720861285, −1.98183774404478459272470304019, −1.74674648865907279351814150715, −1.38693362537108369428351753388, −1.06178736041900295043938235917, −0.55139443671118230987393666572, −0.50395108454835551879734672914, 0.50395108454835551879734672914, 0.55139443671118230987393666572, 1.06178736041900295043938235917, 1.38693362537108369428351753388, 1.74674648865907279351814150715, 1.98183774404478459272470304019, 2.64414493181487465675720861285, 3.07978084709838559390847820032, 3.20721752477552417217794960705, 3.49416434873689011936539627471, 3.72519172146976864084251376904, 4.11312521269399621741835512475, 4.52433254976551109215919417086, 4.73111774103424182475517690557, 4.82477950212275281403964318734, 5.32227055090685237707065931162, 5.47193323223030545607493864514, 5.67890031403640141161490368175, 6.22872659077397790481094885941, 6.44646541365947202295326883839, 6.55141868324495107073112664495, 7.15579270891055384985061994738, 7.28494703873748847247197724141, 7.33926244443408543604324137248, 7.930585679341966779541506241138

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