Properties

Degree 12
Conductor $ 2^{6} \cdot 3^{6} $
Sign $1$
Motivic weight 16
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.00e3·3-s − 9.83e4·4-s − 1.67e5·7-s + 1.29e7·9-s − 5.90e8·12-s + 1.76e9·13-s + 6.44e9·16-s + 6.03e10·19-s − 1.00e9·21-s + 4.19e11·25-s + 1.15e11·27-s + 1.65e10·28-s − 2.84e12·31-s − 1.27e12·36-s + 2.48e12·37-s + 1.05e13·39-s + 4.61e13·43-s + 3.86e13·48-s − 7.86e13·49-s − 1.73e14·52-s + 3.62e14·57-s + 3.06e14·61-s − 2.17e12·63-s − 3.51e14·64-s + 1.97e15·67-s + 3.86e15·73-s + 2.52e15·75-s + ⋯
L(s)  = 1  + 0.915·3-s − 3/2·4-s − 0.0291·7-s + 0.300·9-s − 1.37·12-s + 2.16·13-s + 3/2·16-s + 3.55·19-s − 0.0266·21-s + 2.75·25-s + 0.408·27-s + 0.0436·28-s − 3.33·31-s − 0.450·36-s + 0.707·37-s + 1.97·39-s + 3.94·43-s + 1.37·48-s − 2.36·49-s − 3.24·52-s + 3.25·57-s + 1.59·61-s − 0.00874·63-s − 5/4·64-s + 4.87·67-s + 4.79·73-s + 2.51·75-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+8)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(46656\)    =    \(2^{6} \cdot 3^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(16\)
character  :  induced by $\chi_{6} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(12,\ 46656,\ (\ :[8]^{6}),\ 1)$
$L(\frac{17}{2})$  $\approx$  $11.8276$
$L(\frac12)$  $\approx$  $11.8276$
$L(9)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \(F_p\) is a polynomial of degree 12. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 11.
$p$$F_p$
bad2 \( ( 1 + p^{15} T^{2} )^{3} \)
3 \( 1 - 2002 p T + 285727 p^{4} T^{2} - 2994964 p^{10} T^{3} + 285727 p^{20} T^{4} - 2002 p^{33} T^{5} + p^{48} T^{6} \)
good5 \( 1 - 419902451142 T^{2} + \)\(84\!\cdots\!83\)\( p^{3} T^{4} - \)\(12\!\cdots\!08\)\( p^{6} T^{6} + \)\(84\!\cdots\!83\)\( p^{35} T^{8} - 419902451142 p^{64} T^{10} + p^{96} T^{12} \)
7 \( ( 1 + 83946 T + 5616291552441 p T^{2} + 444833041861625812 p^{3} T^{3} + 5616291552441 p^{17} T^{4} + 83946 p^{32} T^{5} + p^{48} T^{6} )^{2} \)
11 \( 1 - 96714676737722310 T^{2} + \)\(72\!\cdots\!03\)\( p^{2} T^{4} - \)\(28\!\cdots\!20\)\( p^{4} T^{6} + \)\(72\!\cdots\!03\)\( p^{34} T^{8} - 96714676737722310 p^{64} T^{10} + p^{96} T^{12} \)
13 \( ( 1 - 881576070 T + 26023404396976971 p T^{2} + \)\(26\!\cdots\!40\)\( p^{2} T^{3} + 26023404396976971 p^{17} T^{4} - 881576070 p^{32} T^{5} + p^{48} T^{6} )^{2} \)
17 \( 1 - \)\(27\!\cdots\!10\)\( T^{2} + \)\(32\!\cdots\!03\)\( T^{4} - \)\(20\!\cdots\!20\)\( T^{6} + \)\(32\!\cdots\!03\)\( p^{32} T^{8} - \)\(27\!\cdots\!10\)\( p^{64} T^{10} + p^{96} T^{12} \)
19 \( ( 1 - 30153489846 T + \)\(95\!\cdots\!35\)\( T^{2} - \)\(17\!\cdots\!60\)\( T^{3} + \)\(95\!\cdots\!35\)\( p^{16} T^{4} - 30153489846 p^{32} T^{5} + p^{48} T^{6} )^{2} \)
23 \( 1 - \)\(11\!\cdots\!50\)\( T^{2} + \)\(10\!\cdots\!83\)\( T^{4} - \)\(81\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!83\)\( p^{32} T^{8} - \)\(11\!\cdots\!50\)\( p^{64} T^{10} + p^{96} T^{12} \)
29 \( 1 - \)\(39\!\cdots\!06\)\( T^{2} + \)\(19\!\cdots\!35\)\( T^{4} - \)\(47\!\cdots\!00\)\( T^{6} + \)\(19\!\cdots\!35\)\( p^{32} T^{8} - \)\(39\!\cdots\!06\)\( p^{64} T^{10} + p^{96} T^{12} \)
31 \( ( 1 + 45906510486 p T + \)\(24\!\cdots\!95\)\( T^{2} + \)\(19\!\cdots\!40\)\( T^{3} + \)\(24\!\cdots\!95\)\( p^{16} T^{4} + 45906510486 p^{33} T^{5} + p^{48} T^{6} )^{2} \)
37 \( ( 1 - 1241918040966 T + \)\(20\!\cdots\!47\)\( T^{2} - \)\(14\!\cdots\!56\)\( T^{3} + \)\(20\!\cdots\!47\)\( p^{16} T^{4} - 1241918040966 p^{32} T^{5} + p^{48} T^{6} )^{2} \)
41 \( 1 - \)\(74\!\cdots\!50\)\( T^{2} + \)\(65\!\cdots\!23\)\( T^{4} - \)\(54\!\cdots\!00\)\( T^{6} + \)\(65\!\cdots\!23\)\( p^{32} T^{8} - \)\(74\!\cdots\!50\)\( p^{64} T^{10} + p^{96} T^{12} \)
43 \( ( 1 - 23077540595382 T + \)\(57\!\cdots\!79\)\( T^{2} - \)\(66\!\cdots\!76\)\( T^{3} + \)\(57\!\cdots\!79\)\( p^{16} T^{4} - 23077540595382 p^{32} T^{5} + p^{48} T^{6} )^{2} \)
47 \( 1 - \)\(16\!\cdots\!26\)\( T^{2} + \)\(17\!\cdots\!15\)\( T^{4} - \)\(11\!\cdots\!20\)\( T^{6} + \)\(17\!\cdots\!15\)\( p^{32} T^{8} - \)\(16\!\cdots\!26\)\( p^{64} T^{10} + p^{96} T^{12} \)
53 \( 1 - \)\(94\!\cdots\!50\)\( T^{2} + \)\(69\!\cdots\!43\)\( T^{4} - \)\(30\!\cdots\!00\)\( T^{6} + \)\(69\!\cdots\!43\)\( p^{32} T^{8} - \)\(94\!\cdots\!50\)\( p^{64} T^{10} + p^{96} T^{12} \)
59 \( 1 - \)\(64\!\cdots\!50\)\( T^{2} + \)\(27\!\cdots\!63\)\( T^{4} - \)\(68\!\cdots\!00\)\( T^{6} + \)\(27\!\cdots\!63\)\( p^{32} T^{8} - \)\(64\!\cdots\!50\)\( p^{64} T^{10} + p^{96} T^{12} \)
61 \( ( 1 - 153018250949382 T + \)\(11\!\cdots\!71\)\( T^{2} - \)\(11\!\cdots\!08\)\( T^{3} + \)\(11\!\cdots\!71\)\( p^{16} T^{4} - 153018250949382 p^{32} T^{5} + p^{48} T^{6} )^{2} \)
67 \( ( 1 - 989923285004406 T + \)\(60\!\cdots\!47\)\( T^{2} - \)\(25\!\cdots\!36\)\( T^{3} + \)\(60\!\cdots\!47\)\( p^{16} T^{4} - 989923285004406 p^{32} T^{5} + p^{48} T^{6} )^{2} \)
71 \( 1 - \)\(19\!\cdots\!06\)\( T^{2} + \)\(17\!\cdots\!35\)\( T^{4} - \)\(90\!\cdots\!00\)\( T^{6} + \)\(17\!\cdots\!35\)\( p^{32} T^{8} - \)\(19\!\cdots\!06\)\( p^{64} T^{10} + p^{96} T^{12} \)
73 \( ( 1 - 1932103692376710 T + \)\(26\!\cdots\!03\)\( T^{2} - \)\(25\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!03\)\( p^{16} T^{4} - 1932103692376710 p^{32} T^{5} + p^{48} T^{6} )^{2} \)
79 \( ( 1 - 917903242050774 T + \)\(66\!\cdots\!75\)\( T^{2} - \)\(40\!\cdots\!80\)\( T^{3} + \)\(66\!\cdots\!75\)\( p^{16} T^{4} - 917903242050774 p^{32} T^{5} + p^{48} T^{6} )^{2} \)
83 \( 1 - \)\(24\!\cdots\!82\)\( T^{2} + \)\(27\!\cdots\!71\)\( T^{4} - \)\(17\!\cdots\!08\)\( T^{6} + \)\(27\!\cdots\!71\)\( p^{32} T^{8} - \)\(24\!\cdots\!82\)\( p^{64} T^{10} + p^{96} T^{12} \)
89 \( 1 - \)\(58\!\cdots\!70\)\( T^{2} + \)\(17\!\cdots\!83\)\( T^{4} - \)\(34\!\cdots\!40\)\( T^{6} + \)\(17\!\cdots\!83\)\( p^{32} T^{8} - \)\(58\!\cdots\!70\)\( p^{64} T^{10} + p^{96} T^{12} \)
97 \( ( 1 - 15548746562822598 T + \)\(18\!\cdots\!59\)\( T^{2} - \)\(16\!\cdots\!04\)\( T^{3} + \)\(18\!\cdots\!59\)\( p^{16} T^{4} - 15548746562822598 p^{32} T^{5} + p^{48} T^{6} )^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.450931552343509865561183915468, −9.400277279728883639703331128777, −9.353985126093516907070868432293, −8.719216736871403607528844815536, −8.316903820859656618193592963459, −8.294918459689369405650675662991, −7.69620351503980472519863286355, −7.54196060801267272916236767524, −6.99688512224407447495996652940, −6.66888136939713409277169380881, −6.13930717651991674205615488518, −5.60487274385065624811442874071, −5.33064888596798662705154190176, −4.91430216290693060769655079944, −4.89930800873217787276235917993, −3.76998401104160748206688357072, −3.66337044170614071331562990011, −3.63165854064562074591928381867, −3.22223620215909675317754291266, −2.49031371516011271071657554530, −2.20009177237340446336979869047, −1.30961991987416287929297389648, −1.00393931357795407866733308516, −0.74879565564076440136694501021, −0.64693485180959382816645114787, 0.64693485180959382816645114787, 0.74879565564076440136694501021, 1.00393931357795407866733308516, 1.30961991987416287929297389648, 2.20009177237340446336979869047, 2.49031371516011271071657554530, 3.22223620215909675317754291266, 3.63165854064562074591928381867, 3.66337044170614071331562990011, 3.76998401104160748206688357072, 4.89930800873217787276235917993, 4.91430216290693060769655079944, 5.33064888596798662705154190176, 5.60487274385065624811442874071, 6.13930717651991674205615488518, 6.66888136939713409277169380881, 6.99688512224407447495996652940, 7.54196060801267272916236767524, 7.69620351503980472519863286355, 8.294918459689369405650675662991, 8.316903820859656618193592963459, 8.719216736871403607528844815536, 9.353985126093516907070868432293, 9.400277279728883639703331128777, 9.450931552343509865561183915468

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

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