# Properties

 Label 20-370e10-1.1-c1e10-0-1 Degree $20$ Conductor $4.809\times 10^{25}$ Sign $1$ Analytic cond. $50674.3$ Root an. cond. $1.71885$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 10·2-s + 55·4-s − 3·5-s − 220·8-s + 11·9-s + 30·10-s − 2·13-s + 715·16-s + 18·17-s − 110·18-s − 165·20-s + 10·23-s + 7·25-s + 20·26-s − 2.00e3·32-s − 180·34-s + 605·36-s − 8·37-s + 660·40-s − 4·41-s − 10·43-s − 33·45-s − 100·46-s + 31·49-s − 70·50-s − 110·52-s + 5.00e3·64-s + ⋯
 L(s)  = 1 − 7.07·2-s + 55/2·4-s − 1.34·5-s − 77.7·8-s + 11/3·9-s + 9.48·10-s − 0.554·13-s + 178.·16-s + 4.36·17-s − 25.9·18-s − 36.8·20-s + 2.08·23-s + 7/5·25-s + 3.92·26-s − 353.·32-s − 30.8·34-s + 100.·36-s − 1.31·37-s + 104.·40-s − 0.624·41-s − 1.52·43-s − 4.91·45-s − 14.7·46-s + 31/7·49-s − 9.89·50-s − 15.2·52-s + 625.·64-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 5^{10} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 5^{10} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$20$$ Conductor: $$2^{10} \cdot 5^{10} \cdot 37^{10}$$ Sign: $1$ Analytic conductor: $$50674.3$$ Root analytic conductor: $$1.71885$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{370} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(20,\ 2^{10} \cdot 5^{10} \cdot 37^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.1900248400$$ $$L(\frac12)$$ $$\approx$$ $$0.1900248400$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$( 1 + T )^{10}$$
5 $$1 + 3 T + 2 T^{2} - 16 T^{3} - 19 T^{4} - 22 T^{5} - 19 p T^{6} - 16 p^{2} T^{7} + 2 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10}$$
37 $$1 + 8 T + 97 T^{2} + 576 T^{3} + 5282 T^{4} + 25584 T^{5} + 5282 p T^{6} + 576 p^{2} T^{7} + 97 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10}$$
good3 $$1 - 11 T^{2} + 52 T^{4} - 32 p T^{6} - 193 T^{8} + 1390 T^{10} - 193 p^{2} T^{12} - 32 p^{5} T^{14} + 52 p^{6} T^{16} - 11 p^{8} T^{18} + p^{10} T^{20}$$
7 $$1 - 31 T^{2} + 459 T^{4} - 4364 T^{6} + 31924 T^{8} - 218314 T^{10} + 31924 p^{2} T^{12} - 4364 p^{4} T^{14} + 459 p^{6} T^{16} - 31 p^{8} T^{18} + p^{10} T^{20}$$
11 $$( 1 + 27 T^{2} + 51 T^{3} + 302 T^{4} + 1074 T^{5} + 302 p T^{6} + 51 p^{2} T^{7} + 27 p^{3} T^{8} + p^{5} T^{10} )^{2}$$
13 $$( 1 + T + 2 p T^{2} - 48 T^{3} + 329 T^{4} - 1098 T^{5} + 329 p T^{6} - 48 p^{2} T^{7} + 2 p^{4} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2}$$
17 $$( 1 - 9 T + 89 T^{2} - 504 T^{3} + 2982 T^{4} - 12078 T^{5} + 2982 p T^{6} - 504 p^{2} T^{7} + 89 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} )^{2}$$
19 $$1 - 72 T^{2} + 2977 T^{4} - 90248 T^{6} + 2204454 T^{8} - 45625440 T^{10} + 2204454 p^{2} T^{12} - 90248 p^{4} T^{14} + 2977 p^{6} T^{16} - 72 p^{8} T^{18} + p^{10} T^{20}$$
23 $$( 1 - 5 T + 52 T^{2} - 152 T^{3} + 1199 T^{4} - 2470 T^{5} + 1199 p T^{6} - 152 p^{2} T^{7} + 52 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} )^{2}$$
29 $$1 - 108 T^{2} + 5879 T^{4} - 231107 T^{6} + 276700 p T^{8} - 249440874 T^{10} + 276700 p^{3} T^{12} - 231107 p^{4} T^{14} + 5879 p^{6} T^{16} - 108 p^{8} T^{18} + p^{10} T^{20}$$
31 $$1 - 194 T^{2} + 18979 T^{4} - 1222911 T^{6} + 57220962 T^{8} - 2026943118 T^{10} + 57220962 p^{2} T^{12} - 1222911 p^{4} T^{14} + 18979 p^{6} T^{16} - 194 p^{8} T^{18} + p^{10} T^{20}$$
41 $$( 1 + 2 T + 161 T^{2} + 417 T^{3} + 11390 T^{4} + 27434 T^{5} + 11390 p T^{6} + 417 p^{2} T^{7} + 161 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2}$$
43 $$( 1 + 5 T + 127 T^{2} + 784 T^{3} + 8994 T^{4} + 46310 T^{5} + 8994 p T^{6} + 784 p^{2} T^{7} + 127 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} )^{2}$$
47 $$1 - 160 T^{2} + 11993 T^{4} - 511560 T^{6} + 12029318 T^{8} - 253577648 T^{10} + 12029318 p^{2} T^{12} - 511560 p^{4} T^{14} + 11993 p^{6} T^{16} - 160 p^{8} T^{18} + p^{10} T^{20}$$
53 $$1 - 273 T^{2} + 40401 T^{4} - 4128228 T^{6} + 317307718 T^{8} - 18960642934 T^{10} + 317307718 p^{2} T^{12} - 4128228 p^{4} T^{14} + 40401 p^{6} T^{16} - 273 p^{8} T^{18} + p^{10} T^{20}$$
59 $$1 - 352 T^{2} + 63153 T^{4} - 7528424 T^{6} + 659017222 T^{8} - 44169143152 T^{10} + 659017222 p^{2} T^{12} - 7528424 p^{4} T^{14} + 63153 p^{6} T^{16} - 352 p^{8} T^{18} + p^{10} T^{20}$$
61 $$1 - 460 T^{2} + 97647 T^{4} - 12826483 T^{6} + 19411236 p T^{8} - 82282695178 T^{10} + 19411236 p^{3} T^{12} - 12826483 p^{4} T^{14} + 97647 p^{6} T^{16} - 460 p^{8} T^{18} + p^{10} T^{20}$$
67 $$1 - 303 T^{2} + 54020 T^{4} - 6711456 T^{6} + 641201935 T^{8} - 48089295626 T^{10} + 641201935 p^{2} T^{12} - 6711456 p^{4} T^{14} + 54020 p^{6} T^{16} - 303 p^{8} T^{18} + p^{10} T^{20}$$
71 $$( 1 + 10 T + 191 T^{2} + 944 T^{3} + 12662 T^{4} + 37836 T^{5} + 12662 p T^{6} + 944 p^{2} T^{7} + 191 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} )^{2}$$
73 $$1 - 541 T^{2} + 140394 T^{4} - 23094490 T^{6} + 2668314981 T^{8} - 226082874802 T^{10} + 2668314981 p^{2} T^{12} - 23094490 p^{4} T^{14} + 140394 p^{6} T^{16} - 541 p^{8} T^{18} + p^{10} T^{20}$$
79 $$1 - 499 T^{2} + 127952 T^{4} - 21543420 T^{6} + 2613057515 T^{8} - 237224216810 T^{10} + 2613057515 p^{2} T^{12} - 21543420 p^{4} T^{14} + 127952 p^{6} T^{16} - 499 p^{8} T^{18} + p^{10} T^{20}$$
83 $$1 - 364 T^{2} + 62793 T^{4} - 7248824 T^{6} + 680746366 T^{8} - 58266464440 T^{10} + 680746366 p^{2} T^{12} - 7248824 p^{4} T^{14} + 62793 p^{6} T^{16} - 364 p^{8} T^{18} + p^{10} T^{20}$$
89 $$1 - 478 T^{2} + 122989 T^{4} - 21250184 T^{6} + 2738250882 T^{8} - 274126598068 T^{10} + 2738250882 p^{2} T^{12} - 21250184 p^{4} T^{14} + 122989 p^{6} T^{16} - 478 p^{8} T^{18} + p^{10} T^{20}$$
97 $$( 1 - T + 307 T^{2} - 636 T^{3} + 45116 T^{4} - 103398 T^{5} + 45116 p T^{6} - 636 p^{2} T^{7} + 307 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$