# Properties

 Label 16-162e8-1.1-c6e8-0-2 Degree $16$ Conductor $4.744\times 10^{17}$ Sign $1$ Analytic cond. $3.72185\times 10^{12}$ Root an. cond. $6.10481$ Motivic weight $6$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 64·4-s + 4·7-s − 3.83e3·13-s + 1.02e3·16-s + 2.04e4·19-s − 1.84e4·25-s + 256·28-s + 4.00e4·31-s − 4.04e4·37-s − 1.84e5·43-s + 4.70e5·49-s − 2.45e5·52-s − 6.09e5·61-s − 6.55e4·64-s + 2.00e6·67-s + 1.05e6·73-s + 1.31e6·76-s + 8.48e5·79-s − 1.53e4·91-s − 3.62e6·97-s − 1.17e6·100-s − 5.27e6·103-s − 2.48e6·109-s + 4.09e3·112-s − 1.95e6·121-s + 2.56e6·124-s + 127-s + ⋯
 L(s)  = 1 + 4-s + 0.0116·7-s − 1.74·13-s + 1/4·16-s + 2.98·19-s − 1.17·25-s + 0.0116·28-s + 1.34·31-s − 0.797·37-s − 2.32·43-s + 3.99·49-s − 1.74·52-s − 2.68·61-s − 1/4·64-s + 6.67·67-s + 2.70·73-s + 2.98·76-s + 1.72·79-s − 0.0203·91-s − 3.96·97-s − 1.17·100-s − 4.82·103-s − 1.91·109-s + 0.00291·112-s − 1.10·121-s + 1.34·124-s + 0.0348·133-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{8} \cdot 3^{32}$$ Sign: $1$ Analytic conductor: $$3.72185\times 10^{12}$$ Root analytic conductor: $$6.10481$$ Motivic weight: $$6$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{8} \cdot 3^{32} ,\ ( \ : [3]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$16.87080036$$ $$L(\frac12)$$ $$\approx$$ $$16.87080036$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$( 1 - p^{5} T^{2} + p^{10} T^{4} )^{2}$$
3 $$1$$
good5 $$1 + 736 p^{2} T^{2} + 360097 p^{4} T^{4} - 441343136 p^{6} T^{6} - 331209693824 p^{8} T^{8} - 441343136 p^{18} T^{10} + 360097 p^{28} T^{12} + 736 p^{38} T^{14} + p^{48} T^{16}$$
7 $$( 1 - 2 T - 235268 T^{2} + 52 T^{3} + 41511156187 T^{4} + 52 p^{6} T^{5} - 235268 p^{12} T^{6} - 2 p^{18} T^{7} + p^{24} T^{8} )^{2}$$
11 $$1 + 1952680 T^{2} + 3083567918158 T^{4} - 10832424911451056000 T^{6} -$$$$21\!\cdots\!77$$$$T^{8} - 10832424911451056000 p^{12} T^{10} + 3083567918158 p^{24} T^{12} + 1952680 p^{36} T^{14} + p^{48} T^{16}$$
13 $$( 1 + 1918 T - 3356387 T^{2} - 5022296426 T^{3} + 8438203548124 T^{4} - 5022296426 p^{6} T^{5} - 3356387 p^{12} T^{6} + 1918 p^{18} T^{7} + p^{24} T^{8} )^{2}$$
17 $$( 1 - 81233464 T^{2} + 2800570401456999 T^{4} - 81233464 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
19 $$( 1 - 5122 T + 92571840 T^{2} - 5122 p^{6} T^{3} + p^{12} T^{4} )^{4}$$
23 $$1 + 306194656 T^{2} + 45364969486178110 T^{4} +$$$$13\!\cdots\!04$$$$T^{6} -$$$$19\!\cdots\!21$$$$T^{8} +$$$$13\!\cdots\!04$$$$p^{12} T^{10} + 45364969486178110 p^{24} T^{12} + 306194656 p^{36} T^{14} + p^{48} T^{16}$$
29 $$1 - 218801912 T^{2} - 665575825749798599 T^{4} -$$$$12\!\cdots\!32$$$$T^{6} +$$$$36\!\cdots\!64$$$$T^{8} -$$$$12\!\cdots\!32$$$$p^{12} T^{10} - 665575825749798599 p^{24} T^{12} - 218801912 p^{36} T^{14} + p^{48} T^{16}$$
31 $$( 1 - 20048 T - 1257936182 T^{2} + 2308504666048 T^{3} + 1610759885481661411 T^{4} + 2308504666048 p^{6} T^{5} - 1257936182 p^{12} T^{6} - 20048 p^{18} T^{7} + p^{24} T^{8} )^{2}$$
37 $$( 1 + 10100 T + 5013752475 T^{2} + 10100 p^{6} T^{3} + p^{12} T^{4} )^{4}$$
41 $$1 + 2589640888 T^{2} - 11950492095743827346 T^{4} -$$$$68\!\cdots\!36$$$$T^{6} -$$$$31\!\cdots\!25$$$$T^{8} -$$$$68\!\cdots\!36$$$$p^{12} T^{10} - 11950492095743827346 p^{24} T^{12} + 2589640888 p^{36} T^{14} + p^{48} T^{16}$$
43 $$( 1 + 92470 T - 316419596 T^{2} - 349130250016940 T^{3} - 15385889618990334485 T^{4} - 349130250016940 p^{6} T^{5} - 316419596 p^{12} T^{6} + 92470 p^{18} T^{7} + p^{24} T^{8} )^{2}$$
47 $$1 + 27858666964 T^{2} +$$$$35\!\cdots\!42$$$$T^{4} +$$$$24\!\cdots\!12$$$$p^{2} T^{6} +$$$$15\!\cdots\!79$$$$p^{4} T^{8} +$$$$24\!\cdots\!12$$$$p^{14} T^{10} +$$$$35\!\cdots\!42$$$$p^{24} T^{12} + 27858666964 p^{36} T^{14} + p^{48} T^{16}$$
53 $$( 1 - 51925621720 T^{2} +$$$$16\!\cdots\!94$$$$T^{4} - 51925621720 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
59 $$1 + 144480622612 T^{2} +$$$$12\!\cdots\!18$$$$T^{4} +$$$$73\!\cdots\!68$$$$T^{6} +$$$$34\!\cdots\!51$$$$T^{8} +$$$$73\!\cdots\!68$$$$p^{12} T^{10} +$$$$12\!\cdots\!18$$$$p^{24} T^{12} + 144480622612 p^{36} T^{14} + p^{48} T^{16}$$
61 $$( 1 + 304528 T - 5128503047 T^{2} - 1575915008710448 T^{3} +$$$$21\!\cdots\!16$$$$T^{4} - 1575915008710448 p^{6} T^{5} - 5128503047 p^{12} T^{6} + 304528 p^{18} T^{7} + p^{24} T^{8} )^{2}$$
67 $$( 1 - 1004486 T + 576587763532 T^{2} - 252615769683118436 T^{3} +$$$$87\!\cdots\!11$$$$T^{4} - 252615769683118436 p^{6} T^{5} + 576587763532 p^{12} T^{6} - 1004486 p^{18} T^{7} + p^{24} T^{8} )^{2}$$
71 $$( 1 - 94765336960 T^{2} +$$$$38\!\cdots\!74$$$$T^{4} - 94765336960 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
73 $$( 1 - 262912 T + 52515591 p^{2} T^{2} - 262912 p^{6} T^{3} + p^{12} T^{4} )^{4}$$
79 $$( 1 - 424358 T + 116649454564 T^{2} + 179395075645340236 T^{3} -$$$$98\!\cdots\!85$$$$T^{4} + 179395075645340236 p^{6} T^{5} + 116649454564 p^{12} T^{6} - 424358 p^{18} T^{7} + p^{24} T^{8} )^{2}$$
83 $$1 + 413441187988 T^{2} +$$$$11\!\cdots\!58$$$$T^{4} -$$$$64\!\cdots\!68$$$$T^{6} -$$$$26\!\cdots\!09$$$$T^{8} -$$$$64\!\cdots\!68$$$$p^{12} T^{10} +$$$$11\!\cdots\!58$$$$p^{24} T^{12} + 413441187988 p^{36} T^{14} + p^{48} T^{16}$$
89 $$( 1 - 332250007216 T^{2} +$$$$48\!\cdots\!59$$$$T^{4} - 332250007216 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
97 $$( 1 + 1810864 T + 1030883258314 T^{2} + 1054649291167231936 T^{3} +$$$$16\!\cdots\!99$$$$T^{4} + 1054649291167231936 p^{6} T^{5} + 1030883258314 p^{12} T^{6} + 1810864 p^{18} T^{7} + p^{24} T^{8} )^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−4.84613263066478634688569853068, −4.36212220304960894116031636029, −4.24596428752086627774621266271, −4.18627239404344035224339477731, −4.03125914397353907134532791635, −3.74168090690689966546753062254, −3.51229886318853810924615294046, −3.49994927874634060274780474356, −3.45681939014794562178648851978, −3.09359272372174818933377952781, −2.73196538470891222377130174115, −2.71603175419994484376381960300, −2.47406291476136074134517621688, −2.45796904954714194757540929572, −2.27057662456083825419372236583, −2.19924852971607551157292036424, −1.78233249484582640718848110531, −1.45877427017401829768352932781, −1.34918817098490835607613431479, −1.22213026312063924513824729793, −1.17221197598771822830256787170, −0.67716411088750949645448610336, −0.47457504687966383289599796479, −0.39241375109515108283754120049, −0.28564315810198911496395496353, 0.28564315810198911496395496353, 0.39241375109515108283754120049, 0.47457504687966383289599796479, 0.67716411088750949645448610336, 1.17221197598771822830256787170, 1.22213026312063924513824729793, 1.34918817098490835607613431479, 1.45877427017401829768352932781, 1.78233249484582640718848110531, 2.19924852971607551157292036424, 2.27057662456083825419372236583, 2.45796904954714194757540929572, 2.47406291476136074134517621688, 2.71603175419994484376381960300, 2.73196538470891222377130174115, 3.09359272372174818933377952781, 3.45681939014794562178648851978, 3.49994927874634060274780474356, 3.51229886318853810924615294046, 3.74168090690689966546753062254, 4.03125914397353907134532791635, 4.18627239404344035224339477731, 4.24596428752086627774621266271, 4.36212220304960894116031636029, 4.84613263066478634688569853068

## Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.