# Properties

 Label 16-1150e8-1.1-c3e8-0-1 Degree $16$ Conductor $3.059\times 10^{24}$ Sign $1$ Analytic cond. $4.49274\times 10^{14}$ Root an. cond. $8.23724$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 16·4-s + 76·9-s − 78·11-s + 160·16-s − 106·19-s − 322·29-s + 776·31-s − 1.21e3·36-s + 968·41-s + 1.24e3·44-s − 271·49-s + 188·59-s + 2.30e3·61-s − 1.28e3·64-s + 400·71-s + 1.69e3·76-s + 1.81e3·79-s + 1.92e3·81-s + 3.56e3·89-s − 5.92e3·99-s + 732·101-s − 2.80e3·109-s + 5.15e3·116-s − 2.54e3·121-s − 1.24e4·124-s + 127-s + 131-s + ⋯
 L(s)  = 1 − 2·4-s + 2.81·9-s − 2.13·11-s + 5/2·16-s − 1.27·19-s − 2.06·29-s + 4.49·31-s − 5.62·36-s + 3.68·41-s + 4.27·44-s − 0.790·49-s + 0.414·59-s + 4.84·61-s − 5/2·64-s + 0.668·71-s + 2.55·76-s + 2.58·79-s + 2.64·81-s + 4.24·89-s − 6.01·99-s + 0.721·101-s − 2.46·109-s + 4.12·116-s − 1.91·121-s − 8.99·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{8} \cdot 5^{16} \cdot 23^{8}$$ Sign: $1$ Analytic conductor: $$4.49274\times 10^{14}$$ Root analytic conductor: $$8.23724$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1150} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{8} \cdot 5^{16} \cdot 23^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.7620650269$$ $$L(\frac12)$$ $$\approx$$ $$0.7620650269$$ $$L(\frac{5}{2})$$ 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^{2} T^{2} )^{4}$$
5 $$1$$
23 $$( 1 + p^{2} T^{2} )^{4}$$
good3 $$1 - 76 T^{2} + 3848 T^{4} - 152491 T^{6} + 4579960 T^{8} - 152491 p^{6} T^{10} + 3848 p^{12} T^{12} - 76 p^{18} T^{14} + p^{24} T^{16}$$
7 $$1 + 271 T^{2} + 262443 T^{4} + 81521801 T^{6} + 38580527000 T^{8} + 81521801 p^{6} T^{10} + 262443 p^{12} T^{12} + 271 p^{18} T^{14} + p^{24} T^{16}$$
11 $$( 1 + 39 T + 323 p T^{2} + 61307 T^{3} + 4937428 T^{4} + 61307 p^{3} T^{5} + 323 p^{7} T^{6} + 39 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
13 $$1 - 560 p T^{2} + 29010248 T^{4} - 77440358559 T^{6} + 177381917092904 T^{8} - 77440358559 p^{6} T^{10} + 29010248 p^{12} T^{12} - 560 p^{19} T^{14} + p^{24} T^{16}$$
17 $$1 - 10365 T^{2} + 53829207 T^{4} - 325122404275 T^{6} + 1890221243071728 T^{8} - 325122404275 p^{6} T^{10} + 53829207 p^{12} T^{12} - 10365 p^{18} T^{14} + p^{24} T^{16}$$
19 $$( 1 + 53 T + 653 p T^{2} + 750501 T^{3} + 76029128 T^{4} + 750501 p^{3} T^{5} + 653 p^{7} T^{6} + 53 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
29 $$( 1 + 161 T + 2424 p T^{2} + 9893175 T^{3} + 2240848710 T^{4} + 9893175 p^{3} T^{5} + 2424 p^{7} T^{6} + 161 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
31 $$( 1 - 388 T + 132240 T^{2} - 29130015 T^{3} + 5707089166 T^{4} - 29130015 p^{3} T^{5} + 132240 p^{6} T^{6} - 388 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
37 $$1 - 263908 T^{2} + 29678457924 T^{4} - 1997492369236476 T^{6} +$$$$10\!\cdots\!70$$$$T^{8} - 1997492369236476 p^{6} T^{10} + 29678457924 p^{12} T^{12} - 263908 p^{18} T^{14} + p^{24} T^{16}$$
41 $$( 1 - 484 T + 239452 T^{2} - 82501249 T^{3} + 25012433496 T^{4} - 82501249 p^{3} T^{5} + 239452 p^{6} T^{6} - 484 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
43 $$1 - 170324 T^{2} + 12654905140 T^{4} - 56807297462540 T^{6} - 30597519420624769994 T^{8} - 56807297462540 p^{6} T^{10} + 12654905140 p^{12} T^{12} - 170324 p^{18} T^{14} + p^{24} T^{16}$$
47 $$1 - 299579 T^{2} + 49794378038 T^{4} - 7026075724926165 T^{6} +$$$$82\!\cdots\!34$$$$T^{8} - 7026075724926165 p^{6} T^{10} + 49794378038 p^{12} T^{12} - 299579 p^{18} T^{14} + p^{24} T^{16}$$
53 $$1 - 644536 T^{2} + 229405493436 T^{4} - 55168174215712328 T^{6} +$$$$95\!\cdots\!42$$$$T^{8} - 55168174215712328 p^{6} T^{10} + 229405493436 p^{12} T^{12} - 644536 p^{18} T^{14} + p^{24} T^{16}$$
59 $$( 1 - 94 T + 578056 T^{2} - 37319902 T^{3} + 164753482782 T^{4} - 37319902 p^{3} T^{5} + 578056 p^{6} T^{6} - 94 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
61 $$( 1 - 1153 T + 853867 T^{2} - 469738549 T^{3} + 241810092932 T^{4} - 469738549 p^{3} T^{5} + 853867 p^{6} T^{6} - 1153 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
67 $$1 - 1443384 T^{2} + 1011640988028 T^{4} - 463029907725740360 T^{6} +$$$$15\!\cdots\!34$$$$T^{8} - 463029907725740360 p^{6} T^{10} + 1011640988028 p^{12} T^{12} - 1443384 p^{18} T^{14} + p^{24} T^{16}$$
71 $$( 1 - 200 T + 364322 T^{2} - 121468265 T^{3} + 278790401066 T^{4} - 121468265 p^{3} T^{5} + 364322 p^{6} T^{6} - 200 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
73 $$1 - 2508767 T^{2} + 2925586305990 T^{4} - 2073927837042605185 T^{6} +$$$$97\!\cdots\!06$$$$T^{8} - 2073927837042605185 p^{6} T^{10} + 2925586305990 p^{12} T^{12} - 2508767 p^{18} T^{14} + p^{24} T^{16}$$
79 $$( 1 - 908 T + 1048492 T^{2} - 1070426012 T^{3} + 693505308902 T^{4} - 1070426012 p^{3} T^{5} + 1048492 p^{6} T^{6} - 908 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
83 $$1 - 2757104 T^{2} + 4107007404540 T^{4} - 3944325335784523920 T^{6} +$$$$26\!\cdots\!46$$$$T^{8} - 3944325335784523920 p^{6} T^{10} + 4107007404540 p^{12} T^{12} - 2757104 p^{18} T^{14} + p^{24} T^{16}$$
89 $$( 1 - 1784 T + 2327792 T^{2} - 1523808488 T^{3} + 1318662054974 T^{4} - 1523808488 p^{3} T^{5} + 2327792 p^{6} T^{6} - 1784 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
97 $$1 - 1677577 T^{2} + 2970157515891 T^{4} - 3828623340675353139 T^{6} +$$$$36\!\cdots\!28$$$$T^{8} - 3828623340675353139 p^{6} T^{10} + 2970157515891 p^{12} T^{12} - 1677577 p^{18} T^{14} + p^{24} T^{16}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$