Properties

Label 6-405e3-1.1-c3e3-0-1
Degree $6$
Conductor $66430125$
Sign $-1$
Analytic cond. $13644.6$
Root an. cond. $4.88833$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 6·4-s − 15·5-s − 43·7-s − 15·10-s − 14·11-s + 40·13-s − 43·14-s − 16-s + 166·17-s − 164·19-s + 90·20-s − 14·22-s − 171·23-s + 150·25-s + 40·26-s + 258·28-s + 335·29-s − 352·31-s − 71·32-s + 166·34-s + 645·35-s + 402·37-s − 164·38-s − 187·41-s − 602·43-s + 84·44-s + ⋯
L(s)  = 1  + 0.353·2-s − 3/4·4-s − 1.34·5-s − 2.32·7-s − 0.474·10-s − 0.383·11-s + 0.853·13-s − 0.820·14-s − 0.0156·16-s + 2.36·17-s − 1.98·19-s + 1.00·20-s − 0.135·22-s − 1.55·23-s + 6/5·25-s + 0.301·26-s + 1.74·28-s + 2.14·29-s − 2.03·31-s − 0.392·32-s + 0.837·34-s + 3.11·35-s + 1.78·37-s − 0.700·38-s − 0.712·41-s − 2.13·43-s + 0.287·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66430125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66430125 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(66430125\)    =    \(3^{12} \cdot 5^{3}\)
Sign: $-1$
Analytic conductor: \(13644.6\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 66430125,\ (\ :3/2, 3/2, 3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
5$C_1$ \( ( 1 + p T )^{3} \)
good2$S_4\times C_2$ \( 1 - T + 7 T^{2} - 13 T^{3} + 7 p^{3} T^{4} - p^{6} T^{5} + p^{9} T^{6} \)
7$S_4\times C_2$ \( 1 + 43 T + 1224 T^{2} + 24161 T^{3} + 1224 p^{3} T^{4} + 43 p^{6} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 + 14 T + 107 p T^{2} + 80816 T^{3} + 107 p^{4} T^{4} + 14 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 - 40 T + 4139 T^{2} - 100396 T^{3} + 4139 p^{3} T^{4} - 40 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 - 166 T + 23659 T^{2} - 1787440 T^{3} + 23659 p^{3} T^{4} - 166 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 + 164 T + 27869 T^{2} + 2307068 T^{3} + 27869 p^{3} T^{4} + 164 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 + 171 T + 31668 T^{2} + 4099905 T^{3} + 31668 p^{3} T^{4} + 171 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 - 335 T + 100498 T^{2} - 16233563 T^{3} + 100498 p^{3} T^{4} - 335 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 + 352 T + 75441 T^{2} + 11111924 T^{3} + 75441 p^{3} T^{4} + 352 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 - 402 T + 176667 T^{2} - 37389728 T^{3} + 176667 p^{3} T^{4} - 402 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 + 187 T + 162166 T^{2} + 18548179 T^{3} + 162166 p^{3} T^{4} + 187 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 + 14 p T + 248477 T^{2} + 80281904 T^{3} + 248477 p^{3} T^{4} + 14 p^{7} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 + 665 T + 406750 T^{2} + 140572073 T^{3} + 406750 p^{3} T^{4} + 665 p^{6} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 - 730 T + 552931 T^{2} - 220610956 T^{3} + 552931 p^{3} T^{4} - 730 p^{6} T^{5} + p^{9} T^{6} \)
59$S_4\times C_2$ \( 1 - 298 T + 261493 T^{2} + 4970012 T^{3} + 261493 p^{3} T^{4} - 298 p^{6} T^{5} + p^{9} T^{6} \)
61$S_4\times C_2$ \( 1 + 1439 T + 1270250 T^{2} + 708864815 T^{3} + 1270250 p^{3} T^{4} + 1439 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 + 1849 T + 1995966 T^{2} + 1320997733 T^{3} + 1995966 p^{3} T^{4} + 1849 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 + 70 T + 388273 T^{2} - 173667512 T^{3} + 388273 p^{3} T^{4} + 70 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 + 368 T + 794123 T^{2} + 151388768 T^{3} + 794123 p^{3} T^{4} + 368 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 + 382 T + 1091193 T^{2} + 238359212 T^{3} + 1091193 p^{3} T^{4} + 382 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 - 831 T + 554388 T^{2} + 5533827 T^{3} + 554388 p^{3} T^{4} - 831 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 + 1719 T + 2353398 T^{2} + 2298177027 T^{3} + 2353398 p^{3} T^{4} + 1719 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 + 282 T + 2670351 T^{2} + 497849308 T^{3} + 2670351 p^{3} T^{4} + 282 p^{6} T^{5} + p^{9} T^{6} \)
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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

−10.24875642006252447952241018384, −9.836635628737163923941086191976, −9.387497099677342209012454848898, −9.313060015961307458530779984313, −8.796635924682996660342435748979, −8.478251245375619225219958585680, −8.241468857387486694705658157683, −7.87973241201408964873032836549, −7.75465425815915971386712795979, −7.26410626473961432051573184844, −6.78177424502527471420094530003, −6.49285035912593536973233502239, −6.32313918493435378312590957439, −5.99450815449592522591125190951, −5.42975786565682355835803925075, −5.30101055555994870108317503134, −4.54268929610437647346207249766, −4.25270629603758556462606798602, −4.13358198422322150890978338145, −3.60584747587702006637721215153, −3.19899183299458647403895862166, −3.10334985191545992703641541796, −2.62641628492093770026139391178, −1.58636584409372252049226466524, −1.23438658473595018547536285964, 0, 0, 0, 1.23438658473595018547536285964, 1.58636584409372252049226466524, 2.62641628492093770026139391178, 3.10334985191545992703641541796, 3.19899183299458647403895862166, 3.60584747587702006637721215153, 4.13358198422322150890978338145, 4.25270629603758556462606798602, 4.54268929610437647346207249766, 5.30101055555994870108317503134, 5.42975786565682355835803925075, 5.99450815449592522591125190951, 6.32313918493435378312590957439, 6.49285035912593536973233502239, 6.78177424502527471420094530003, 7.26410626473961432051573184844, 7.75465425815915971386712795979, 7.87973241201408964873032836549, 8.241468857387486694705658157683, 8.478251245375619225219958585680, 8.796635924682996660342435748979, 9.313060015961307458530779984313, 9.387497099677342209012454848898, 9.836635628737163923941086191976, 10.24875642006252447952241018384

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