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Properties

Label 12-75e6-1.1-c17e6-0-2
Degree $12$
Conductor $177978515625$
Sign $1$
Analytic cond. $6.73340\times 10^{12}$
Root an. cond. $11.7224$
Motivic weight $17$
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  + 4.00e5·4-s − 1.29e8·9-s + 1.88e9·11-s + 6.96e10·16-s + 1.56e11·19-s − 1.15e13·29-s + 1.11e13·31-s − 5.16e13·36-s + 3.34e14·41-s + 7.55e14·44-s + 6.28e14·49-s − 9.31e14·59-s − 4.63e15·61-s + 5.74e15·64-s + 1.29e16·71-s + 6.25e16·76-s − 1.74e16·79-s + 1.11e16·81-s + 3.44e16·89-s − 2.43e17·99-s − 2.10e16·101-s + 7.76e17·109-s − 4.62e18·116-s + 2.17e17·121-s + 4.45e18·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 3.05·4-s − 9-s + 2.65·11-s + 4.05·16-s + 2.11·19-s − 4.28·29-s + 2.34·31-s − 3.05·36-s + 6.55·41-s + 8.10·44-s + 2.70·49-s − 0.825·59-s − 3.09·61-s + 2.55·64-s + 2.37·71-s + 6.44·76-s − 1.29·79-s + 2/3·81-s + 0.927·89-s − 2.65·99-s − 0.193·101-s + 3.73·109-s − 13.0·116-s + 0.430·121-s + 7.15·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(18-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+17/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{12}\)
Sign: $1$
Analytic conductor: \(6.73340\times 10^{12}\)
Root analytic conductor: \(11.7224\)
Motivic weight: \(17\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 5^{12} ,\ ( \ : [17/2]^{6} ),\ 1 )\)

Particular Values

\(L(9)\) \(\approx\) \(85.78442565\)
\(L(\frac12)\) \(\approx\) \(85.78442565\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + p^{16} T^{2} )^{3} \)
5 \( 1 \)
good2 \( 1 - 400303 T^{2} + 5664295497 p^{4} T^{4} - 3456914821799 p^{12} T^{6} + 5664295497 p^{38} T^{8} - 400303 p^{68} T^{10} + p^{102} T^{12} \)
7 \( 1 - 628921209447674 T^{2} + \)\(90\!\cdots\!87\)\( p^{4} T^{4} - \)\(92\!\cdots\!52\)\( p^{8} T^{6} + \)\(90\!\cdots\!87\)\( p^{38} T^{8} - 628921209447674 p^{68} T^{10} + p^{102} T^{12} \)
11 \( ( 1 - 943563680 T + 111526088429338691 p T^{2} - \)\(59\!\cdots\!44\)\( p^{2} T^{3} + 111526088429338691 p^{18} T^{4} - 943563680 p^{34} T^{5} + p^{51} T^{6} )^{2} \)
13 \( 1 - 15666233794155858 p^{2} T^{2} + \)\(69\!\cdots\!47\)\( p^{4} T^{4} - \)\(70\!\cdots\!84\)\( p^{6} T^{6} + \)\(69\!\cdots\!47\)\( p^{38} T^{8} - 15666233794155858 p^{70} T^{10} + p^{102} T^{12} \)
17 \( 1 - \)\(66\!\cdots\!30\)\( T^{2} + \)\(63\!\cdots\!87\)\( T^{4} - \)\(12\!\cdots\!40\)\( T^{6} + \)\(63\!\cdots\!87\)\( p^{34} T^{8} - \)\(66\!\cdots\!30\)\( p^{68} T^{10} + p^{102} T^{12} \)
19 \( ( 1 - 78122492996 T + \)\(17\!\cdots\!77\)\( T^{2} - \)\(84\!\cdots\!88\)\( T^{3} + \)\(17\!\cdots\!77\)\( p^{17} T^{4} - 78122492996 p^{34} T^{5} + p^{51} T^{6} )^{2} \)
23 \( 1 - \)\(52\!\cdots\!26\)\( T^{2} + \)\(13\!\cdots\!47\)\( T^{4} - \)\(24\!\cdots\!68\)\( T^{6} + \)\(13\!\cdots\!47\)\( p^{34} T^{8} - \)\(52\!\cdots\!26\)\( p^{68} T^{10} + p^{102} T^{12} \)
29 \( ( 1 + 5775268588078 T + \)\(31\!\cdots\!87\)\( T^{2} + \)\(88\!\cdots\!04\)\( T^{3} + \)\(31\!\cdots\!87\)\( p^{17} T^{4} + 5775268588078 p^{34} T^{5} + p^{51} T^{6} )^{2} \)
31 \( ( 1 - 5565463149104 T + \)\(58\!\cdots\!37\)\( T^{2} - \)\(19\!\cdots\!88\)\( T^{3} + \)\(58\!\cdots\!37\)\( p^{17} T^{4} - 5565463149104 p^{34} T^{5} + p^{51} T^{6} )^{2} \)
37 \( 1 - \)\(11\!\cdots\!74\)\( T^{2} + \)\(91\!\cdots\!27\)\( T^{4} - \)\(45\!\cdots\!72\)\( T^{6} + \)\(91\!\cdots\!27\)\( p^{34} T^{8} - \)\(11\!\cdots\!74\)\( p^{68} T^{10} + p^{102} T^{12} \)
41 \( ( 1 - 167461457288254 T + \)\(15\!\cdots\!27\)\( T^{2} - \)\(93\!\cdots\!08\)\( T^{3} + \)\(15\!\cdots\!27\)\( p^{17} T^{4} - 167461457288254 p^{34} T^{5} + p^{51} T^{6} )^{2} \)
43 \( 1 + \)\(18\!\cdots\!90\)\( T^{2} + \)\(20\!\cdots\!47\)\( T^{4} + \)\(14\!\cdots\!20\)\( T^{6} + \)\(20\!\cdots\!47\)\( p^{34} T^{8} + \)\(18\!\cdots\!90\)\( p^{68} T^{10} + p^{102} T^{12} \)
47 \( 1 - \)\(35\!\cdots\!10\)\( T^{2} + \)\(15\!\cdots\!07\)\( T^{4} - \)\(52\!\cdots\!80\)\( T^{6} + \)\(15\!\cdots\!07\)\( p^{34} T^{8} - \)\(35\!\cdots\!10\)\( p^{68} T^{10} + p^{102} T^{12} \)
53 \( 1 + \)\(39\!\cdots\!94\)\( T^{2} + \)\(11\!\cdots\!87\)\( T^{4} - \)\(32\!\cdots\!28\)\( T^{6} + \)\(11\!\cdots\!87\)\( p^{34} T^{8} + \)\(39\!\cdots\!94\)\( p^{68} T^{10} + p^{102} T^{12} \)
59 \( ( 1 + 465601947196256 T + \)\(32\!\cdots\!37\)\( T^{2} + \)\(11\!\cdots\!28\)\( T^{3} + \)\(32\!\cdots\!37\)\( p^{17} T^{4} + 465601947196256 p^{34} T^{5} + p^{51} T^{6} )^{2} \)
61 \( ( 1 + 2317809676510478 T + \)\(76\!\cdots\!19\)\( T^{2} + \)\(10\!\cdots\!44\)\( T^{3} + \)\(76\!\cdots\!19\)\( p^{17} T^{4} + 2317809676510478 p^{34} T^{5} + p^{51} T^{6} )^{2} \)
67 \( 1 - \)\(51\!\cdots\!90\)\( T^{2} + \)\(12\!\cdots\!87\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!87\)\( p^{34} T^{8} - \)\(51\!\cdots\!90\)\( p^{68} T^{10} + p^{102} T^{12} \)
71 \( ( 1 - 6465608483990656 T + \)\(10\!\cdots\!85\)\( T^{2} - \)\(38\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!85\)\( p^{17} T^{4} - 6465608483990656 p^{34} T^{5} + p^{51} T^{6} )^{2} \)
73 \( 1 - \)\(96\!\cdots\!06\)\( T^{2} + \)\(54\!\cdots\!27\)\( T^{4} - \)\(28\!\cdots\!08\)\( T^{6} + \)\(54\!\cdots\!27\)\( p^{34} T^{8} - \)\(96\!\cdots\!06\)\( p^{68} T^{10} + p^{102} T^{12} \)
79 \( ( 1 + 8740508940658880 T + \)\(43\!\cdots\!77\)\( T^{2} + \)\(31\!\cdots\!40\)\( T^{3} + \)\(43\!\cdots\!77\)\( p^{17} T^{4} + 8740508940658880 p^{34} T^{5} + p^{51} T^{6} )^{2} \)
83 \( 1 - \)\(97\!\cdots\!22\)\( T^{2} + \)\(55\!\cdots\!07\)\( T^{4} - \)\(22\!\cdots\!76\)\( T^{6} + \)\(55\!\cdots\!07\)\( p^{34} T^{8} - \)\(97\!\cdots\!22\)\( p^{68} T^{10} + p^{102} T^{12} \)
89 \( ( 1 - 17217755358726426 T + \)\(12\!\cdots\!87\)\( T^{2} + \)\(30\!\cdots\!92\)\( T^{3} + \)\(12\!\cdots\!87\)\( p^{17} T^{4} - 17217755358726426 p^{34} T^{5} + p^{51} T^{6} )^{2} \)
97 \( 1 - \)\(18\!\cdots\!90\)\( T^{2} + \)\(22\!\cdots\!07\)\( T^{4} - \)\(15\!\cdots\!20\)\( T^{6} + \)\(22\!\cdots\!07\)\( p^{34} T^{8} - \)\(18\!\cdots\!90\)\( p^{68} T^{10} + p^{102} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.31378524632494569743909085458, −4.93359487275777056436062158175, −4.52431489574096805464574809499, −4.52010662412983167637659687028, −4.24270725711736619058191562741, −3.84604204301378339362943209623, −3.83702804165021191562603866511, −3.73279905764233663821752044190, −3.37057697364086095170876987251, −3.30767126327739125256275581772, −2.92943511503006041710005270472, −2.65583871197156876998159255470, −2.53803765802789144191743955701, −2.47930619717334067538254912113, −2.28169363605382948290548847971, −2.21455340608177936535070440047, −1.76601857712670105248704617387, −1.52166807932030689912084403833, −1.44620741513241312774020215961, −1.24650497383917139468158563852, −1.01718394903131148071660950046, −0.984488203538739914833421419143, −0.52152055728567989124971167506, −0.43724592555007185778693894566, −0.36448101604039340291851399190, 0.36448101604039340291851399190, 0.43724592555007185778693894566, 0.52152055728567989124971167506, 0.984488203538739914833421419143, 1.01718394903131148071660950046, 1.24650497383917139468158563852, 1.44620741513241312774020215961, 1.52166807932030689912084403833, 1.76601857712670105248704617387, 2.21455340608177936535070440047, 2.28169363605382948290548847971, 2.47930619717334067538254912113, 2.53803765802789144191743955701, 2.65583871197156876998159255470, 2.92943511503006041710005270472, 3.30767126327739125256275581772, 3.37057697364086095170876987251, 3.73279905764233663821752044190, 3.83702804165021191562603866511, 3.84604204301378339362943209623, 4.24270725711736619058191562741, 4.52010662412983167637659687028, 4.52431489574096805464574809499, 4.93359487275777056436062158175, 5.31378524632494569743909085458

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

Plot not available for L-functions of degree greater than 10.