Properties

 Degree 8 Conductor $3^{4}$ Sign $1$ Motivic weight 16 Primitive no Self-dual yes Analytic rank 0

Origins of factors

Dirichlet series

 L(s)  = 1 − 2.05e3·3-s + 1.24e5·4-s − 3.14e6·7-s + 1.14e7·9-s − 2.56e8·12-s − 1.58e9·13-s + 3.94e9·16-s − 5.61e10·19-s + 6.44e9·21-s + 3.01e11·25-s − 1.26e11·27-s − 3.92e11·28-s + 2.47e12·31-s + 1.42e12·36-s + 3.70e11·37-s + 3.24e12·39-s − 2.80e13·43-s − 8.10e12·48-s − 4.67e13·49-s − 1.97e14·52-s + 1.15e14·57-s + 3.62e14·61-s − 3.58e13·63-s − 4.23e14·64-s − 2.67e13·67-s + 3.17e14·73-s − 6.17e14·75-s + ⋯
 L(s)  = 1 − 0.312·3-s + 1.90·4-s − 0.544·7-s + 0.265·9-s − 0.595·12-s − 1.93·13-s + 0.919·16-s − 3.30·19-s + 0.170·21-s + 1.97·25-s − 0.448·27-s − 1.03·28-s + 2.89·31-s + 0.505·36-s + 0.105·37-s + 0.606·39-s − 2.40·43-s − 0.287·48-s − 1.40·49-s − 3.69·52-s + 1.03·57-s + 1.88·61-s − 0.144·63-s − 1.50·64-s − 0.0659·67-s + 0.393·73-s − 0.617·75-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+8)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$8$$ $$N$$ = $$81$$    =    $$3^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$16$$ character : induced by $\chi_{3} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(8,\ 81,\ (\ :8, 8, 8, 8),\ 1)$ $L(\frac{17}{2})$ $\approx$ $1.83003$ $L(\frac12)$ $\approx$ $1.83003$ $L(9)$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 3$, $$F_p$$ is a polynomial of degree 8. If $p = 3$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad3$D_{4}$ $$1 + 76 p^{3} T - 122 p^{10} T^{2} + 76 p^{19} T^{3} + p^{32} T^{4}$$
good2$C_2^2 \wr C_2$ $$1 - 15605 p^{3} T^{2} + 11364297 p^{10} T^{4} - 15605 p^{35} T^{6} + p^{64} T^{8}$$
5$C_2^2 \wr C_2$ $$1 - 12045550276 p^{2} T^{2} + 20208485552534687982 p^{5} T^{4} - 12045550276 p^{34} T^{6} + p^{64} T^{8}$$
7$D_{4}$ $$( 1 + 224396 p T + 79000079514 p^{3} T^{2} + 224396 p^{17} T^{3} + p^{32} T^{4} )^{2}$$
11$C_2^2 \wr C_2$ $$1 - 5004710841163244 p T^{2} +$$$$15\!\cdots\!26$$$$p^{3} T^{4} - 5004710841163244 p^{33} T^{6} + p^{64} T^{8}$$
13$D_{4}$ $$( 1 + 60797324 p T + 8193950692930518 p^{2} T^{2} + 60797324 p^{17} T^{3} + p^{32} T^{4} )^{2}$$
17$C_2^2 \wr C_2$ $$1 -$$$$13\!\cdots\!60$$$$T^{2} +$$$$81\!\cdots\!58$$$$T^{4} -$$$$13\!\cdots\!60$$$$p^{32} T^{6} + p^{64} T^{8}$$
19$D_{4}$ $$( 1 + 1476766220 p T +$$$$42\!\cdots\!98$$$$T^{2} + 1476766220 p^{17} T^{3} + p^{32} T^{4} )^{2}$$
23$C_2^2 \wr C_2$ $$1 -$$$$90\!\cdots\!80$$$$p T^{2} +$$$$18\!\cdots\!78$$$$T^{4} -$$$$90\!\cdots\!80$$$$p^{33} T^{6} + p^{64} T^{8}$$
29$C_2^2 \wr C_2$ $$1 -$$$$48\!\cdots\!44$$$$T^{2} +$$$$17\!\cdots\!66$$$$T^{4} -$$$$48\!\cdots\!44$$$$p^{32} T^{6} + p^{64} T^{8}$$
31$D_{4}$ $$( 1 - 39867438004 p T +$$$$18\!\cdots\!06$$$$T^{2} - 39867438004 p^{17} T^{3} + p^{32} T^{4} )^{2}$$
37$D_{4}$ $$( 1 - 185281606948 T +$$$$24\!\cdots\!22$$$$T^{2} - 185281606948 p^{16} T^{3} + p^{32} T^{4} )^{2}$$
41$C_2^2 \wr C_2$ $$1 -$$$$23\!\cdots\!04$$$$T^{2} +$$$$21\!\cdots\!66$$$$T^{4} -$$$$23\!\cdots\!04$$$$p^{32} T^{6} + p^{64} T^{8}$$
43$D_{4}$ $$( 1 + 14032511031332 T +$$$$22\!\cdots\!02$$$$T^{2} + 14032511031332 p^{16} T^{3} + p^{32} T^{4} )^{2}$$
47$C_2^2 \wr C_2$ $$1 -$$$$18\!\cdots\!60$$$$T^{2} +$$$$14\!\cdots\!98$$$$T^{4} -$$$$18\!\cdots\!60$$$$p^{32} T^{6} + p^{64} T^{8}$$
53$C_2^2 \wr C_2$ $$1 -$$$$12\!\cdots\!60$$$$T^{2} +$$$$71\!\cdots\!98$$$$T^{4} -$$$$12\!\cdots\!60$$$$p^{32} T^{6} + p^{64} T^{8}$$
59$C_2^2 \wr C_2$ $$1 -$$$$16\!\cdots\!04$$$$T^{2} +$$$$93\!\cdots\!66$$$$T^{4} -$$$$16\!\cdots\!04$$$$p^{32} T^{6} + p^{64} T^{8}$$
61$D_{4}$ $$( 1 - 181134896541604 T +$$$$81\!\cdots\!26$$$$T^{2} - 181134896541604 p^{16} T^{3} + p^{32} T^{4} )^{2}$$
67$D_{4}$ $$( 1 + 13387202681732 T +$$$$30\!\cdots\!42$$$$T^{2} + 13387202681732 p^{16} T^{3} + p^{32} T^{4} )^{2}$$
71$C_2^2 \wr C_2$ $$1 -$$$$12\!\cdots\!44$$$$T^{2} +$$$$66\!\cdots\!66$$$$T^{4} -$$$$12\!\cdots\!44$$$$p^{32} T^{6} + p^{64} T^{8}$$
73$D_{4}$ $$( 1 - 158814443769988 T +$$$$37\!\cdots\!62$$$$T^{2} - 158814443769988 p^{16} T^{3} + p^{32} T^{4} )^{2}$$
79$D_{4}$ $$( 1 - 2397008582592460 T +$$$$59\!\cdots\!22$$$$p T^{2} - 2397008582592460 p^{16} T^{3} + p^{32} T^{4} )^{2}$$
83$C_2^2 \wr C_2$ $$1 -$$$$46\!\cdots\!60$$$$T^{2} -$$$$18\!\cdots\!42$$$$T^{4} -$$$$46\!\cdots\!60$$$$p^{32} T^{6} + p^{64} T^{8}$$
89$C_2^2 \wr C_2$ $$1 -$$$$45\!\cdots\!84$$$$T^{2} +$$$$22\!\cdots\!06$$$$T^{4} -$$$$45\!\cdots\!84$$$$p^{32} T^{6} + p^{64} T^{8}$$
97$D_{4}$ $$( 1 + 6916897501300892 T +$$$$13\!\cdots\!62$$$$T^{2} + 6916897501300892 p^{16} T^{3} + p^{32} T^{4} )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}