# Properties

 Degree 4 Conductor $2^{4}$ Sign $1$ Motivic weight 17 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 5.88e3·3-s + 6.04e5·5-s + 2.53e7·7-s + 1.12e8·9-s + 1.25e9·11-s − 1.32e9·13-s − 3.55e9·15-s − 2.74e10·17-s + 1.01e11·19-s − 1.49e11·21-s + 1.34e11·23-s − 8.04e11·25-s − 1.88e12·27-s + 2.33e12·29-s + 2.78e11·31-s − 7.40e12·33-s + 1.53e13·35-s + 2.09e13·37-s + 7.76e12·39-s + 4.16e12·41-s − 1.11e14·43-s + 6.80e13·45-s + 1.96e14·47-s + 1.38e14·49-s + 1.61e14·51-s − 4.87e14·53-s + 7.60e14·55-s + ⋯
 L(s)  = 1 − 0.517·3-s + 0.691·5-s + 1.66·7-s + 0.872·9-s + 1.77·11-s − 0.448·13-s − 0.357·15-s − 0.956·17-s + 1.36·19-s − 0.859·21-s + 0.358·23-s − 1.05·25-s − 1.28·27-s + 0.867·29-s + 0.0587·31-s − 0.916·33-s + 1.14·35-s + 0.979·37-s + 0.232·39-s + 0.0814·41-s − 1.45·43-s + 0.603·45-s + 1.20·47-s + 0.595·49-s + 0.494·51-s − 1.07·53-s + 1.22·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\n
\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$16$$    =    $$2^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$17$$ character : induced by $\chi_{4} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 16,\ (\ :17/2, 17/2),\ 1)$ $L(9)$ $\approx$ $2.98068$ $L(\frac12)$ $\approx$ $2.98068$ $L(\frac{19}{2})$ 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$D_{4}$ $$1 + 1960 p T - 321646 p^{5} T^{2} + 1960 p^{18} T^{3} + p^{34} T^{4}$$
5$D_{4}$ $$1 - 604044 T + 46794698206 p^{2} T^{2} - 604044 p^{17} T^{3} + p^{34} T^{4}$$
7$D_{4}$ $$1 - 25350160 T + 10288992992430 p^{2} T^{2} - 25350160 p^{17} T^{3} + p^{34} T^{4}$$
11$D_{4}$ $$1 - 114513480 p T + 7489675618956502 p^{2} T^{2} - 114513480 p^{18} T^{3} + p^{34} T^{4}$$
13$D_{4}$ $$1 + 1320052580 T + 661181623540177494 p T^{2} + 1320052580 p^{17} T^{3} + p^{34} T^{4}$$
17$D_{4}$ $$1 + 27498226140 T +$$$$18\!\cdots\!58$$$$T^{2} + 27498226140 p^{17} T^{3} + p^{34} T^{4}$$
19$D_{4}$ $$1 - 101133633832 T +$$$$10\!\cdots\!34$$$$T^{2} - 101133633832 p^{17} T^{3} + p^{34} T^{4}$$
23$D_{4}$ $$1 - 134767491120 T +$$$$21\!\cdots\!70$$$$T^{2} - 134767491120 p^{17} T^{3} + p^{34} T^{4}$$
29$D_{4}$ $$1 - 2337155582652 T +$$$$94\!\cdots\!94$$$$T^{2} - 2337155582652 p^{17} T^{3} + p^{34} T^{4}$$
31$D_{4}$ $$1 - 278836113472 T +$$$$39\!\cdots\!18$$$$T^{2} - 278836113472 p^{17} T^{3} + p^{34} T^{4}$$
37$D_{4}$ $$1 - 20929802888140 T +$$$$93\!\cdots\!10$$$$T^{2} - 20929802888140 p^{17} T^{3} + p^{34} T^{4}$$
41$D_{4}$ $$1 - 4166592315732 T +$$$$39\!\cdots\!18$$$$T^{2} - 4166592315732 p^{17} T^{3} + p^{34} T^{4}$$
43$D_{4}$ $$1 + 111143148534440 T +$$$$13\!\cdots\!86$$$$T^{2} + 111143148534440 p^{17} T^{3} + p^{34} T^{4}$$
47$D_{4}$ $$1 - 196772651157600 T +$$$$53\!\cdots\!18$$$$T^{2} - 196772651157600 p^{17} T^{3} + p^{34} T^{4}$$
53$D_{4}$ $$1 + 487965122736660 T +$$$$41\!\cdots\!50$$$$T^{2} + 487965122736660 p^{17} T^{3} + p^{34} T^{4}$$
59$D_{4}$ $$1 - 2835904197813624 T +$$$$45\!\cdots\!82$$$$T^{2} - 2835904197813624 p^{17} T^{3} + p^{34} T^{4}$$
61$D_{4}$ $$1 + 67544034994436 T -$$$$59\!\cdots\!34$$$$T^{2} + 67544034994436 p^{17} T^{3} + p^{34} T^{4}$$
67$D_{4}$ $$1 + 995171321546360 T +$$$$18\!\cdots\!70$$$$T^{2} + 995171321546360 p^{17} T^{3} + p^{34} T^{4}$$
71$D_{4}$ $$1 - 3882245493215376 T +$$$$21\!\cdots\!26$$$$T^{2} - 3882245493215376 p^{17} T^{3} + p^{34} T^{4}$$
73$D_{4}$ $$1 + 12746425881769580 T +$$$$13\!\cdots\!82$$$$T^{2} + 12746425881769580 p^{17} T^{3} + p^{34} T^{4}$$
79$D_{4}$ $$1 + 14984271534065504 T +$$$$37\!\cdots\!22$$$$T^{2} + 14984271534065504 p^{17} T^{3} + p^{34} T^{4}$$
83$D_{4}$ $$1 - 43899417809893800 T +$$$$13\!\cdots\!30$$$$T^{2} - 43899417809893800 p^{17} T^{3} + p^{34} T^{4}$$
89$D_{4}$ $$1 - 51909007958846388 T +$$$$34\!\cdots\!94$$$$T^{2} - 51909007958846388 p^{17} T^{3} + p^{34} T^{4}$$
97$D_{4}$ $$1 + 49281007789848380 T +$$$$91\!\cdots\!10$$$$T^{2} + 49281007789848380 p^{17} T^{3} + p^{34} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}