# Properties

 Label 4-2e8-1.1-c21e2-0-1 Degree $4$ Conductor $256$ Sign $1$ Analytic cond. $1999.55$ Root an. cond. $6.68703$ Motivic weight $21$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 1.05e5·3-s + 2.10e6·5-s − 4.44e8·7-s + 6.33e8·9-s − 5.38e10·11-s − 4.90e11·13-s + 2.22e11·15-s − 6.59e12·17-s − 1.93e13·19-s − 4.68e13·21-s + 4.09e14·23-s − 9.45e14·25-s + 6.45e13·27-s − 2.40e15·29-s + 8.68e15·31-s − 5.67e15·33-s − 9.37e14·35-s − 2.18e15·37-s − 5.17e16·39-s + 6.81e16·41-s − 2.64e17·43-s + 1.33e15·45-s − 4.26e17·47-s − 7.33e17·49-s − 6.95e17·51-s − 3.05e18·53-s − 1.13e17·55-s + ⋯
 L(s)  = 1 + 1.03·3-s + 0.0965·5-s − 0.595·7-s + 0.0605·9-s − 0.625·11-s − 0.986·13-s + 0.0995·15-s − 0.793·17-s − 0.722·19-s − 0.613·21-s + 2.06·23-s − 1.98·25-s + 0.0603·27-s − 1.06·29-s + 1.90·31-s − 0.644·33-s − 0.0574·35-s − 0.0747·37-s − 1.01·39-s + 0.793·41-s − 1.86·43-s + 0.00584·45-s − 1.18·47-s − 1.31·49-s − 0.817·51-s − 2.40·53-s − 0.0603·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$256$$    =    $$2^{8}$$ Sign: $1$ Analytic conductor: $$1999.55$$ Root analytic conductor: $$6.68703$$ Motivic weight: $$21$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 256,\ (\ :21/2, 21/2),\ 1)$$

## Particular Values

 $$L(11)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{23}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
good3$D_{4}$ $$1 - 35144 p T + 129409046 p^{4} T^{2} - 35144 p^{22} T^{3} + p^{42} T^{4}$$
5$D_{4}$ $$1 - 421628 p T + 7595986878694 p^{3} T^{2} - 421628 p^{22} T^{3} + p^{42} T^{4}$$
7$D_{4}$ $$1 + 444771792 T + 132991518054685058 p T^{2} + 444771792 p^{21} T^{3} + p^{42} T^{4}$$
11$D_{4}$ $$1 + 53806403320 T + 97941017461487903606 p^{2} T^{2} + 53806403320 p^{21} T^{3} + p^{42} T^{4}$$
13$D_{4}$ $$1 + 490366676932 T +$$$$23\!\cdots\!06$$$$p T^{2} + 490366676932 p^{21} T^{3} + p^{42} T^{4}$$
17$D_{4}$ $$1 + 387874392476 p T +$$$$47\!\cdots\!06$$$$p^{2} T^{2} + 387874392476 p^{22} T^{3} + p^{42} T^{4}$$
19$D_{4}$ $$1 + 1015915680280 p T -$$$$69\!\cdots\!18$$$$p^{2} T^{2} + 1015915680280 p^{22} T^{3} + p^{42} T^{4}$$
23$D_{4}$ $$1 - 409737865776272 T +$$$$10\!\cdots\!18$$$$T^{2} - 409737865776272 p^{21} T^{3} + p^{42} T^{4}$$
29$D_{4}$ $$1 + 2404787522145060 T +$$$$46\!\cdots\!74$$$$T^{2} + 2404787522145060 p^{21} T^{3} + p^{42} T^{4}$$
31$D_{4}$ $$1 - 8689907170559168 T +$$$$56\!\cdots\!82$$$$T^{2} - 8689907170559168 p^{21} T^{3} + p^{42} T^{4}$$
37$D_{4}$ $$1 + 2186204096251860 T +$$$$16\!\cdots\!58$$$$T^{2} + 2186204096251860 p^{21} T^{3} + p^{42} T^{4}$$
41$D_{4}$ $$1 - 68178038573558676 T +$$$$15\!\cdots\!90$$$$T^{2} - 68178038573558676 p^{21} T^{3} + p^{42} T^{4}$$
43$D_{4}$ $$1 + 264529652266004024 T +$$$$49\!\cdots\!34$$$$T^{2} + 264529652266004024 p^{21} T^{3} + p^{42} T^{4}$$
47$D_{4}$ $$1 + 426494411558622432 T +$$$$23\!\cdots\!50$$$$T^{2} + 426494411558622432 p^{21} T^{3} + p^{42} T^{4}$$
53$D_{4}$ $$1 + 3055980275589518132 T +$$$$54\!\cdots\!62$$$$T^{2} + 3055980275589518132 p^{21} T^{3} + p^{42} T^{4}$$
59$D_{4}$ $$1 + 783424997522814424 T +$$$$30\!\cdots\!06$$$$T^{2} + 783424997522814424 p^{21} T^{3} + p^{42} T^{4}$$
61$D_{4}$ $$1 + 7177279049078597092 T +$$$$67\!\cdots\!62$$$$T^{2} + 7177279049078597092 p^{21} T^{3} + p^{42} T^{4}$$
67$D_{4}$ $$1 + 16674123174011538088 T +$$$$43\!\cdots\!06$$$$T^{2} + 16674123174011538088 p^{21} T^{3} + p^{42} T^{4}$$
71$D_{4}$ $$1 + 9448263149848716368 T +$$$$82\!\cdots\!02$$$$T^{2} + 9448263149848716368 p^{21} T^{3} + p^{42} T^{4}$$
73$D_{4}$ $$1 + 11586140334503007532 T -$$$$95\!\cdots\!02$$$$T^{2} + 11586140334503007532 p^{21} T^{3} + p^{42} T^{4}$$
79$D_{4}$ $$1 - 85280702218715897824 T +$$$$14\!\cdots\!38$$$$T^{2} - 85280702218715897824 p^{21} T^{3} + p^{42} T^{4}$$
83$D_{4}$ $$1 -$$$$38\!\cdots\!16$$$$T +$$$$71\!\cdots\!14$$$$T^{2} -$$$$38\!\cdots\!16$$$$p^{21} T^{3} + p^{42} T^{4}$$
89$D_{4}$ $$1 + 59742932430695979660 T +$$$$15\!\cdots\!22$$$$T^{2} + 59742932430695979660 p^{21} T^{3} + p^{42} T^{4}$$
97$D_{4}$ $$1 -$$$$78\!\cdots\!88$$$$T +$$$$12\!\cdots\!30$$$$T^{2} -$$$$78\!\cdots\!88$$$$p^{21} T^{3} + p^{42} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$