# Properties

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

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## 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.