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Properties

Label 6-2e18-1.1-c23e3-0-0
Degree $6$
Conductor $262144$
Sign $1$
Analytic cond. $9.87342\times 10^{6}$
Root an. cond. $14.6468$
Motivic weight $23$
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  − 3.27e4·3-s − 3.14e7·5-s + 9.93e8·7-s − 1.39e11·9-s + 2.34e10·11-s − 2.01e12·13-s + 1.02e12·15-s − 2.16e12·17-s − 3.12e14·19-s − 3.24e13·21-s + 4.77e15·23-s − 7.48e15·25-s − 1.44e16·27-s − 1.72e17·29-s + 4.24e17·31-s − 7.66e14·33-s − 3.12e16·35-s − 2.54e18·37-s + 6.60e16·39-s + 2.76e18·41-s + 5.96e18·43-s + 4.40e18·45-s − 5.25e19·47-s − 5.86e19·49-s + 7.06e16·51-s − 1.56e19·53-s − 7.37e17·55-s + ⋯
L(s)  = 1  − 0.106·3-s − 0.288·5-s + 0.189·7-s − 1.48·9-s + 0.0247·11-s − 0.312·13-s + 0.0307·15-s − 0.0152·17-s − 0.614·19-s − 0.0202·21-s + 1.04·23-s − 0.627·25-s − 0.499·27-s − 2.62·29-s + 3.00·31-s − 0.00264·33-s − 0.0547·35-s − 2.35·37-s + 0.0333·39-s + 0.783·41-s + 0.978·43-s + 0.428·45-s − 3.10·47-s − 2.14·49-s + 0.00162·51-s − 0.232·53-s − 0.00714·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+23/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(262144\)    =    \(2^{18}\)
Sign: $1$
Analytic conductor: \(9.87342\times 10^{6}\)
Root analytic conductor: \(14.6468\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 262144,\ (\ :23/2, 23/2, 23/2),\ 1)\)

Particular Values

\(L(12)\) \(\approx\) \(0.05204098015\)
\(L(\frac12)\) \(\approx\) \(0.05204098015\)
\(L(\frac{25}{2})\) 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$S_4\times C_2$ \( 1 + 32708 T + 15672779593 p^{2} T^{2} + 10806152860616 p^{7} T^{3} + 15672779593 p^{25} T^{4} + 32708 p^{46} T^{5} + p^{69} T^{6} \)
5$S_4\times C_2$ \( 1 + 1259226 p^{2} T + 338935636667019 p^{2} T^{2} + \)\(42\!\cdots\!64\)\( p^{5} T^{3} + 338935636667019 p^{25} T^{4} + 1259226 p^{48} T^{5} + p^{69} T^{6} \)
7$S_4\times C_2$ \( 1 - 141860760 p T + 1217330227180247349 p^{2} T^{2} - \)\(14\!\cdots\!48\)\( p^{4} T^{3} + 1217330227180247349 p^{25} T^{4} - 141860760 p^{47} T^{5} + p^{69} T^{6} \)
11$S_4\times C_2$ \( 1 - 2131047804 p T + \)\(51\!\cdots\!07\)\( p T^{2} + \)\(38\!\cdots\!56\)\( p^{2} T^{3} + \)\(51\!\cdots\!07\)\( p^{24} T^{4} - 2131047804 p^{47} T^{5} + p^{69} T^{6} \)
13$S_4\times C_2$ \( 1 + 2019379246962 T + \)\(33\!\cdots\!07\)\( p T^{2} + \)\(43\!\cdots\!88\)\( p^{2} T^{3} + \)\(33\!\cdots\!07\)\( p^{24} T^{4} + 2019379246962 p^{46} T^{5} + p^{69} T^{6} \)
17$S_4\times C_2$ \( 1 + 2160517821354 T + \)\(13\!\cdots\!27\)\( p T^{2} + \)\(51\!\cdots\!36\)\( p^{2} T^{3} + \)\(13\!\cdots\!27\)\( p^{24} T^{4} + 2160517821354 p^{46} T^{5} + p^{69} T^{6} \)
19$S_4\times C_2$ \( 1 + 312191787410964 T + \)\(17\!\cdots\!79\)\( p T^{2} - \)\(38\!\cdots\!92\)\( p^{2} T^{3} + \)\(17\!\cdots\!79\)\( p^{24} T^{4} + 312191787410964 p^{46} T^{5} + p^{69} T^{6} \)
23$S_4\times C_2$ \( 1 - 4776103684514040 T + \)\(31\!\cdots\!13\)\( T^{2} - \)\(19\!\cdots\!08\)\( T^{3} + \)\(31\!\cdots\!13\)\( p^{23} T^{4} - 4776103684514040 p^{46} T^{5} + p^{69} T^{6} \)
29$S_4\times C_2$ \( 1 + 172150256810092098 T + \)\(20\!\cdots\!03\)\( T^{2} + \)\(15\!\cdots\!72\)\( T^{3} + \)\(20\!\cdots\!03\)\( p^{23} T^{4} + 172150256810092098 p^{46} T^{5} + p^{69} T^{6} \)
31$S_4\times C_2$ \( 1 - 424442363135920032 T + \)\(11\!\cdots\!53\)\( T^{2} - \)\(19\!\cdots\!24\)\( T^{3} + \)\(11\!\cdots\!53\)\( p^{23} T^{4} - 424442363135920032 p^{46} T^{5} + p^{69} T^{6} \)
37$S_4\times C_2$ \( 1 + 2543857445080432362 T + \)\(52\!\cdots\!95\)\( T^{2} + \)\(61\!\cdots\!04\)\( T^{3} + \)\(52\!\cdots\!95\)\( p^{23} T^{4} + 2543857445080432362 p^{46} T^{5} + p^{69} T^{6} \)
41$S_4\times C_2$ \( 1 - 2761653172426159758 T + \)\(33\!\cdots\!83\)\( T^{2} - \)\(69\!\cdots\!52\)\( T^{3} + \)\(33\!\cdots\!83\)\( p^{23} T^{4} - 2761653172426159758 p^{46} T^{5} + p^{69} T^{6} \)
43$S_4\times C_2$ \( 1 - 5964473505832923060 T + \)\(41\!\cdots\!89\)\( T^{2} - \)\(49\!\cdots\!44\)\( T^{3} + \)\(41\!\cdots\!89\)\( p^{23} T^{4} - 5964473505832923060 p^{46} T^{5} + p^{69} T^{6} \)
47$S_4\times C_2$ \( 1 + 52578892233811303824 T + \)\(17\!\cdots\!61\)\( T^{2} + \)\(34\!\cdots\!16\)\( T^{3} + \)\(17\!\cdots\!61\)\( p^{23} T^{4} + 52578892233811303824 p^{46} T^{5} + p^{69} T^{6} \)
53$S_4\times C_2$ \( 1 + 15683594666449490106 T + \)\(67\!\cdots\!43\)\( T^{2} - \)\(11\!\cdots\!68\)\( T^{3} + \)\(67\!\cdots\!43\)\( p^{23} T^{4} + 15683594666449490106 p^{46} T^{5} + p^{69} T^{6} \)
59$S_4\times C_2$ \( 1 - \)\(10\!\cdots\!52\)\( T - \)\(31\!\cdots\!23\)\( T^{2} + \)\(36\!\cdots\!80\)\( T^{3} - \)\(31\!\cdots\!23\)\( p^{23} T^{4} - \)\(10\!\cdots\!52\)\( p^{46} T^{5} + p^{69} T^{6} \)
61$S_4\times C_2$ \( 1 + \)\(12\!\cdots\!98\)\( T + \)\(84\!\cdots\!83\)\( T^{2} + \)\(34\!\cdots\!76\)\( T^{3} + \)\(84\!\cdots\!83\)\( p^{23} T^{4} + \)\(12\!\cdots\!98\)\( p^{46} T^{5} + p^{69} T^{6} \)
67$S_4\times C_2$ \( 1 - \)\(80\!\cdots\!12\)\( T + \)\(20\!\cdots\!49\)\( T^{2} - \)\(18\!\cdots\!36\)\( T^{3} + \)\(20\!\cdots\!49\)\( p^{23} T^{4} - \)\(80\!\cdots\!12\)\( p^{46} T^{5} + p^{69} T^{6} \)
71$S_4\times C_2$ \( 1 + \)\(32\!\cdots\!24\)\( T + \)\(11\!\cdots\!57\)\( T^{2} + \)\(20\!\cdots\!88\)\( T^{3} + \)\(11\!\cdots\!57\)\( p^{23} T^{4} + \)\(32\!\cdots\!24\)\( p^{46} T^{5} + p^{69} T^{6} \)
73$S_4\times C_2$ \( 1 - \)\(25\!\cdots\!30\)\( T + \)\(22\!\cdots\!23\)\( T^{2} - \)\(35\!\cdots\!88\)\( T^{3} + \)\(22\!\cdots\!23\)\( p^{23} T^{4} - \)\(25\!\cdots\!30\)\( p^{46} T^{5} + p^{69} T^{6} \)
79$S_4\times C_2$ \( 1 + \)\(17\!\cdots\!12\)\( T + \)\(20\!\cdots\!93\)\( T^{2} + \)\(16\!\cdots\!76\)\( T^{3} + \)\(20\!\cdots\!93\)\( p^{23} T^{4} + \)\(17\!\cdots\!12\)\( p^{46} T^{5} + p^{69} T^{6} \)
83$S_4\times C_2$ \( 1 - \)\(15\!\cdots\!04\)\( T + \)\(38\!\cdots\!41\)\( T^{2} - \)\(42\!\cdots\!48\)\( T^{3} + \)\(38\!\cdots\!41\)\( p^{23} T^{4} - \)\(15\!\cdots\!04\)\( p^{46} T^{5} + p^{69} T^{6} \)
89$S_4\times C_2$ \( 1 + \)\(13\!\cdots\!06\)\( T - \)\(37\!\cdots\!89\)\( T^{2} - \)\(22\!\cdots\!48\)\( T^{3} - \)\(37\!\cdots\!89\)\( p^{23} T^{4} + \)\(13\!\cdots\!06\)\( p^{46} T^{5} + p^{69} T^{6} \)
97$S_4\times C_2$ \( 1 + \)\(12\!\cdots\!66\)\( T + \)\(98\!\cdots\!71\)\( T^{2} - \)\(23\!\cdots\!16\)\( T^{3} + \)\(98\!\cdots\!71\)\( p^{23} T^{4} + \)\(12\!\cdots\!66\)\( p^{46} T^{5} + p^{69} T^{6} \)
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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

−9.345204427664644365841890257571, −8.799024221528586227064290377064, −8.469964998621577486518606399854, −8.234452198346074032564150007809, −7.79813787332106674040570864950, −7.48906071529783151281292292141, −7.16676351764302070216435139681, −6.55091414205524099922967137100, −6.19756539149563458517740011507, −6.11075874511094559906377326811, −5.55156382274342296091545785224, −5.18650844897429872276529125870, −4.82071114480327893019583838448, −4.56043402444213153077677689352, −4.07237777567061352286681576976, −3.62886259382546418988883993901, −3.07206583310598665996340952690, −3.01480932833622032853319404565, −2.77985238397191224564085117580, −1.84248036700612153862678437219, −1.80616013794139910353774206394, −1.62639369491093593778938463867, −0.855969042453938478598309234132, −0.43028771646727902755255772682, −0.04343594764978290103445316007, 0.04343594764978290103445316007, 0.43028771646727902755255772682, 0.855969042453938478598309234132, 1.62639369491093593778938463867, 1.80616013794139910353774206394, 1.84248036700612153862678437219, 2.77985238397191224564085117580, 3.01480932833622032853319404565, 3.07206583310598665996340952690, 3.62886259382546418988883993901, 4.07237777567061352286681576976, 4.56043402444213153077677689352, 4.82071114480327893019583838448, 5.18650844897429872276529125870, 5.55156382274342296091545785224, 6.11075874511094559906377326811, 6.19756539149563458517740011507, 6.55091414205524099922967137100, 7.16676351764302070216435139681, 7.48906071529783151281292292141, 7.79813787332106674040570864950, 8.234452198346074032564150007809, 8.469964998621577486518606399854, 8.799024221528586227064290377064, 9.345204427664644365841890257571

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