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Properties

Label 6-2e18-1.1-c11e3-0-0
Degree $6$
Conductor $262144$
Sign $1$
Analytic cond. $118906.$
Root an. cond. $7.01241$
Motivic weight $11$
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  + 440·3-s − 770·5-s + 6.03e3·7-s − 6.30e4·9-s + 2.36e5·11-s − 1.43e6·13-s − 3.38e5·15-s − 7.56e5·17-s + 2.32e7·19-s + 2.65e6·21-s − 4.23e7·23-s − 7.64e7·25-s + 1.73e7·27-s − 4.26e7·29-s − 5.55e8·31-s + 1.03e8·33-s − 4.64e6·35-s + 4.62e8·37-s − 6.31e8·39-s + 3.57e8·41-s + 1.40e9·43-s + 4.85e7·45-s − 1.62e9·47-s − 1.16e9·49-s − 3.32e8·51-s + 5.08e9·53-s − 1.81e8·55-s + ⋯
L(s)  = 1  + 1.04·3-s − 0.110·5-s + 0.135·7-s − 0.356·9-s + 0.442·11-s − 1.07·13-s − 0.115·15-s − 0.129·17-s + 2.15·19-s + 0.141·21-s − 1.37·23-s − 1.56·25-s + 0.233·27-s − 0.386·29-s − 3.48·31-s + 0.462·33-s − 0.0149·35-s + 1.09·37-s − 1.12·39-s + 0.482·41-s + 1.45·43-s + 0.0392·45-s − 1.03·47-s − 0.591·49-s − 0.135·51-s + 1.67·53-s − 0.0487·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+11/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(262144\)    =    \(2^{18}\)
Sign: $1$
Analytic conductor: \(118906.\)
Root analytic conductor: \(7.01241\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 262144,\ (\ :11/2, 11/2, 11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(0.08665010818\)
\(L(\frac12)\) \(\approx\) \(0.08665010818\)
\(L(\frac{13}{2})\) 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 \( 1 \)
good3$S_4\times C_2$ \( 1 - 440 T + 3169 p^{4} T^{2} - 1951696 p^{4} T^{3} + 3169 p^{15} T^{4} - 440 p^{22} T^{5} + p^{33} T^{6} \)
5$S_4\times C_2$ \( 1 + 154 p T + 15399079 p T^{2} + 8965400556 p^{2} T^{3} + 15399079 p^{12} T^{4} + 154 p^{23} T^{5} + p^{33} T^{6} \)
7$S_4\times C_2$ \( 1 - 6032 T + 172146131 p T^{2} + 2268139380256 p^{2} T^{3} + 172146131 p^{12} T^{4} - 6032 p^{22} T^{5} + p^{33} T^{6} \)
11$S_4\times C_2$ \( 1 - 236136 T + 658806197433 T^{2} - 94106697292238192 T^{3} + 658806197433 p^{11} T^{4} - 236136 p^{22} T^{5} + p^{33} T^{6} \)
13$S_4\times C_2$ \( 1 + 1435722 T + 1658742133611 T^{2} + 317386190851454428 T^{3} + 1658742133611 p^{11} T^{4} + 1435722 p^{22} T^{5} + p^{33} T^{6} \)
17$S_4\times C_2$ \( 1 + 756186 T + 90876740279199 T^{2} + 49784077728432970476 T^{3} + 90876740279199 p^{11} T^{4} + 756186 p^{22} T^{5} + p^{33} T^{6} \)
19$S_4\times C_2$ \( 1 - 23267992 T + 354606805051457 T^{2} - \)\(36\!\cdots\!96\)\( T^{3} + 354606805051457 p^{11} T^{4} - 23267992 p^{22} T^{5} + p^{33} T^{6} \)
23$S_4\times C_2$ \( 1 + 42366288 T + 2006139822787797 T^{2} + \)\(54\!\cdots\!40\)\( T^{3} + 2006139822787797 p^{11} T^{4} + 42366288 p^{22} T^{5} + p^{33} T^{6} \)
29$S_4\times C_2$ \( 1 + 42667514 T + 15098503535225147 T^{2} + \)\(14\!\cdots\!84\)\( T^{3} + 15098503535225147 p^{11} T^{4} + 42667514 p^{22} T^{5} + p^{33} T^{6} \)
31$S_4\times C_2$ \( 1 + 555856064 T + 166006563086475293 T^{2} + \)\(32\!\cdots\!68\)\( T^{3} + 166006563086475293 p^{11} T^{4} + 555856064 p^{22} T^{5} + p^{33} T^{6} \)
37$S_4\times C_2$ \( 1 - 462651966 T + 325547165600951139 T^{2} - \)\(71\!\cdots\!16\)\( T^{3} + 325547165600951139 p^{11} T^{4} - 462651966 p^{22} T^{5} + p^{33} T^{6} \)
41$S_4\times C_2$ \( 1 - 357860750 T + 932688087906818903 T^{2} - \)\(56\!\cdots\!00\)\( T^{3} + 932688087906818903 p^{11} T^{4} - 357860750 p^{22} T^{5} + p^{33} T^{6} \)
43$S_4\times C_2$ \( 1 - 1400057768 T + 2131100485074928601 T^{2} - \)\(26\!\cdots\!08\)\( T^{3} + 2131100485074928601 p^{11} T^{4} - 1400057768 p^{22} T^{5} + p^{33} T^{6} \)
47$S_4\times C_2$ \( 1 + 1620045984 T + 5456072006294623533 T^{2} + \)\(73\!\cdots\!60\)\( T^{3} + 5456072006294623533 p^{11} T^{4} + 1620045984 p^{22} T^{5} + p^{33} T^{6} \)
53$S_4\times C_2$ \( 1 - 5087465774 T + 31922158971563680691 T^{2} - \)\(89\!\cdots\!56\)\( T^{3} + 31922158971563680691 p^{11} T^{4} - 5087465774 p^{22} T^{5} + p^{33} T^{6} \)
59$S_4\times C_2$ \( 1 + 13552245240 T + \)\(14\!\cdots\!77\)\( T^{2} + \)\(89\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!77\)\( p^{11} T^{4} + 13552245240 p^{22} T^{5} + p^{33} T^{6} \)
61$S_4\times C_2$ \( 1 - 18288485958 T + \)\(24\!\cdots\!03\)\( T^{2} - \)\(18\!\cdots\!36\)\( T^{3} + \)\(24\!\cdots\!03\)\( p^{11} T^{4} - 18288485958 p^{22} T^{5} + p^{33} T^{6} \)
67$S_4\times C_2$ \( 1 + 17471404360 T + \)\(23\!\cdots\!81\)\( T^{2} + \)\(17\!\cdots\!92\)\( T^{3} + \)\(23\!\cdots\!81\)\( p^{11} T^{4} + 17471404360 p^{22} T^{5} + p^{33} T^{6} \)
71$S_4\times C_2$ \( 1 - 23838346512 T + \)\(45\!\cdots\!13\)\( T^{2} - \)\(68\!\cdots\!04\)\( T^{3} + \)\(45\!\cdots\!13\)\( p^{11} T^{4} - 23838346512 p^{22} T^{5} + p^{33} T^{6} \)
73$S_4\times C_2$ \( 1 - 35602893150 T + \)\(10\!\cdots\!31\)\( T^{2} - \)\(22\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!31\)\( p^{11} T^{4} - 35602893150 p^{22} T^{5} + p^{33} T^{6} \)
79$S_4\times C_2$ \( 1 + 4634961248 T + \)\(18\!\cdots\!37\)\( T^{2} + \)\(55\!\cdots\!84\)\( T^{3} + \)\(18\!\cdots\!37\)\( p^{11} T^{4} + 4634961248 p^{22} T^{5} + p^{33} T^{6} \)
83$S_4\times C_2$ \( 1 - 49374840984 T + \)\(44\!\cdots\!45\)\( T^{2} - \)\(12\!\cdots\!76\)\( T^{3} + \)\(44\!\cdots\!45\)\( p^{11} T^{4} - 49374840984 p^{22} T^{5} + p^{33} T^{6} \)
89$S_4\times C_2$ \( 1 - 58640505454 T + \)\(65\!\cdots\!11\)\( T^{2} - \)\(33\!\cdots\!88\)\( T^{3} + \)\(65\!\cdots\!11\)\( p^{11} T^{4} - 58640505454 p^{22} T^{5} + p^{33} T^{6} \)
97$S_4\times C_2$ \( 1 + 66696660810 T + \)\(44\!\cdots\!59\)\( T^{2} - \)\(31\!\cdots\!40\)\( T^{3} + \)\(44\!\cdots\!59\)\( p^{11} T^{4} + 66696660810 p^{22} T^{5} + p^{33} T^{6} \)
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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

−11.32423801781153363700619950896, −10.61206668684533078849447070889, −10.34044065467091522085321744721, −9.744063550354984333090766004589, −9.261754272574371000001834241967, −9.187918997142437856556822968603, −9.131177995963953077837633578584, −7.972745064084119541425849063826, −7.917096669717986911935158606420, −7.84217499497691901799653306623, −7.21170676434957158493324503393, −6.79078499756846260758468997761, −6.15751444008090617667210959271, −5.65524040434355688555804866658, −5.29792806804817858101507515559, −4.98599922015355334989876603935, −4.09138852284600699686069756287, −3.64729739203379484171054435065, −3.63068026200975392299896069169, −2.78597724525825656438034892987, −2.36649934264133940990469107099, −2.09882077474103701356301493261, −1.41132640470592417077808487694, −0.893098279024108973771600288932, −0.04567560693418030997657125931, 0.04567560693418030997657125931, 0.893098279024108973771600288932, 1.41132640470592417077808487694, 2.09882077474103701356301493261, 2.36649934264133940990469107099, 2.78597724525825656438034892987, 3.63068026200975392299896069169, 3.64729739203379484171054435065, 4.09138852284600699686069756287, 4.98599922015355334989876603935, 5.29792806804817858101507515559, 5.65524040434355688555804866658, 6.15751444008090617667210959271, 6.79078499756846260758468997761, 7.21170676434957158493324503393, 7.84217499497691901799653306623, 7.917096669717986911935158606420, 7.972745064084119541425849063826, 9.131177995963953077837633578584, 9.187918997142437856556822968603, 9.261754272574371000001834241967, 9.744063550354984333090766004589, 10.34044065467091522085321744721, 10.61206668684533078849447070889, 11.32423801781153363700619950896

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