# Properties

 Degree 4 Conductor $2^{6}$ Sign $1$ Motivic weight 19 Primitive no Self-dual yes Analytic rank 2

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2.79e4·3-s + 1.22e6·5-s + 8.85e7·7-s − 4.01e8·9-s − 7.16e9·11-s − 1.01e10·13-s − 3.42e10·15-s − 7.20e10·17-s − 3.12e12·19-s − 2.47e12·21-s − 1.47e13·23-s − 3.44e13·25-s + 1.16e13·27-s − 3.02e13·29-s − 1.23e14·31-s + 1.99e14·33-s + 1.08e14·35-s + 2.01e15·37-s + 2.82e14·39-s + 2.54e15·41-s − 5.63e15·43-s − 4.92e14·45-s − 2.19e16·47-s − 3.29e15·49-s + 2.01e15·51-s − 9.41e15·53-s − 8.78e15·55-s + ⋯
 L(s)  = 1 − 0.818·3-s + 0.280·5-s + 0.829·7-s − 0.345·9-s − 0.916·11-s − 0.264·13-s − 0.229·15-s − 0.147·17-s − 2.21·19-s − 0.678·21-s − 1.70·23-s − 1.80·25-s + 0.295·27-s − 0.387·29-s − 0.838·31-s + 0.749·33-s + 0.232·35-s + 2.54·37-s + 0.216·39-s + 1.21·41-s − 1.70·43-s − 0.0969·45-s − 2.86·47-s − 0.289·49-s + 0.120·51-s − 0.392·53-s − 0.257·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+19/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$64$$    =    $$2^{6}$$ $$\varepsilon$$ = $1$ motivic weight = $$19$$ character : induced by $\chi_{8} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$2$$ Selberg data = $$(4,\ 64,\ (\ :19/2, 19/2),\ 1)$$ $$L(10)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{21}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 2$,$$F_p(T)$$ is a polynomial of degree 4. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
good3$D_{4}$ $$1 + 9304 p T + 14570470 p^{4} T^{2} + 9304 p^{20} T^{3} + p^{38} T^{4}$$
5$D_{4}$ $$1 - 245324 p T + 1437226145582 p^{2} T^{2} - 245324 p^{20} T^{3} + p^{38} T^{4}$$
7$D_{4}$ $$1 - 88510512 T + 1589951031584546 p T^{2} - 88510512 p^{19} T^{3} + p^{38} T^{4}$$
11$D_{4}$ $$1 + 7163787608 T + 6207054254616739618 p T^{2} + 7163787608 p^{19} T^{3} + p^{38} T^{4}$$
13$D_{4}$ $$1 + 10126923604 T +$$$$22\!\cdots\!66$$$$p T^{2} + 10126923604 p^{19} T^{3} + p^{38} T^{4}$$
17$D_{4}$ $$1 + 4237945820 p T +$$$$26\!\cdots\!54$$$$p^{2} T^{2} + 4237945820 p^{20} T^{3} + p^{38} T^{4}$$
19$D_{4}$ $$1 + 164235814328 p T +$$$$54\!\cdots\!14$$$$T^{2} + 164235814328 p^{20} T^{3} + p^{38} T^{4}$$
23$D_{4}$ $$1 + 14759207090288 T +$$$$68\!\cdots\!70$$$$p T^{2} + 14759207090288 p^{19} T^{3} + p^{38} T^{4}$$
29$D_{4}$ $$1 + 30249539245044 T +$$$$11\!\cdots\!22$$$$T^{2} + 30249539245044 p^{19} T^{3} + p^{38} T^{4}$$
31$D_{4}$ $$1 + 123389562777920 T +$$$$27\!\cdots\!42$$$$T^{2} + 123389562777920 p^{19} T^{3} + p^{38} T^{4}$$
37$D_{4}$ $$1 - 2015393170174524 T +$$$$60\!\cdots\!70$$$$p T^{2} - 2015393170174524 p^{19} T^{3} + p^{38} T^{4}$$
41$D_{4}$ $$1 - 2540784959504244 T +$$$$74\!\cdots\!06$$$$T^{2} - 2540784959504244 p^{19} T^{3} + p^{38} T^{4}$$
43$D_{4}$ $$1 + 5633655093389464 T +$$$$29\!\cdots\!38$$$$T^{2} + 5633655093389464 p^{19} T^{3} + p^{38} T^{4}$$
47$D_{4}$ $$1 + 21948339587130336 T +$$$$23\!\cdots\!90$$$$T^{2} + 21948339587130336 p^{19} T^{3} + p^{38} T^{4}$$
53$D_{4}$ $$1 + 9418125066904676 T +$$$$29\!\cdots\!78$$$$T^{2} + 9418125066904676 p^{19} T^{3} + p^{38} T^{4}$$
59$D_{4}$ $$1 - 98542449590407624 T +$$$$95\!\cdots\!22$$$$T^{2} - 98542449590407624 p^{19} T^{3} + p^{38} T^{4}$$
61$D_{4}$ $$1 - 10292145377839820 T +$$$$11\!\cdots\!82$$$$T^{2} - 10292145377839820 p^{19} T^{3} + p^{38} T^{4}$$
67$D_{4}$ $$1 - 75753628003984504 T +$$$$44\!\cdots\!10$$$$T^{2} - 75753628003984504 p^{19} T^{3} + p^{38} T^{4}$$
71$D_{4}$ $$1 - 17407052566713776 T +$$$$27\!\cdots\!06$$$$T^{2} - 17407052566713776 p^{19} T^{3} + p^{38} T^{4}$$
73$D_{4}$ $$1 + 857508255059832268 T +$$$$53\!\cdots\!30$$$$T^{2} + 857508255059832268 p^{19} T^{3} + p^{38} T^{4}$$
79$D_{4}$ $$1 + 226291921444855072 T +$$$$21\!\cdots\!34$$$$T^{2} + 226291921444855072 p^{19} T^{3} + p^{38} T^{4}$$
83$D_{4}$ $$1 - 767515701460985048 T +$$$$59\!\cdots\!70$$$$T^{2} - 767515701460985048 p^{19} T^{3} + p^{38} T^{4}$$
89$D_{4}$ $$1 - 6092545894435174548 T +$$$$31\!\cdots\!94$$$$T^{2} - 6092545894435174548 p^{19} T^{3} + p^{38} T^{4}$$
97$D_{4}$ $$1 - 1548148249522347076 T +$$$$66\!\cdots\!10$$$$T^{2} - 1548148249522347076 p^{19} T^{3} + p^{38} T^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}