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

Origins of factors

Downloads

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Normalization:  

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 \( ( 1 - T + 307 T^{2} - 636 T^{3} + 45116 T^{4} - 103398 T^{5} + 45116 p T^{6} - 636 p^{2} T^{7} + 307 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.17905450228621559789044926836, −4.05493135743222481555777632216, −4.04688652980808904983934830242, −3.93717686973128500184503330128, −3.66003666680008734498482460014, −3.64250923049939807967731791641, −3.34950857922547723215366095390, −3.25735311500383785121737984036, −3.12463151612735575852697414868, −3.09165077297633279601414564889, −2.87184479737424485641990223304, −2.80636339663185185620768448702, −2.68347517565279136380534429680, −2.39325109486815813608967620118, −2.18381892287677649437099199069, −1.90476899800056057084679979235, −1.80734505940661399190237106758, −1.79878270560593480113151527628, −1.61057951853498497082538792134, −1.20856738834502712407637389645, −1.15202431365447672837557696624, −1.09669338376445786986416008131, −0.986373115977527113288308057356, −0.66658622304901902266149138920, −0.39426706405684723631154924903, 0.39426706405684723631154924903, 0.66658622304901902266149138920, 0.986373115977527113288308057356, 1.09669338376445786986416008131, 1.15202431365447672837557696624, 1.20856738834502712407637389645, 1.61057951853498497082538792134, 1.79878270560593480113151527628, 1.80734505940661399190237106758, 1.90476899800056057084679979235, 2.18381892287677649437099199069, 2.39325109486815813608967620118, 2.68347517565279136380534429680, 2.80636339663185185620768448702, 2.87184479737424485641990223304, 3.09165077297633279601414564889, 3.12463151612735575852697414868, 3.25735311500383785121737984036, 3.34950857922547723215366095390, 3.64250923049939807967731791641, 3.66003666680008734498482460014, 3.93717686973128500184503330128, 4.04688652980808904983934830242, 4.05493135743222481555777632216, 4.17905450228621559789044926836

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

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