Properties

Label 20-420e10-1.1-c1e10-0-0
Degree $20$
Conductor $1.708\times 10^{26}$
Sign $1$
Analytic cond. $179996.$
Root an. cond. $1.83131$
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  + 3-s + 5·5-s − 5·7-s + 2·9-s + 6·11-s + 5·15-s − 6·17-s + 3·19-s − 5·21-s − 24·23-s + 10·25-s − 27-s + 15·31-s + 6·33-s − 25·35-s − 37-s + 8·41-s − 26·43-s + 10·45-s − 14·47-s + 6·49-s − 6·51-s + 24·53-s + 30·55-s + 3·57-s + 42·61-s − 10·63-s + ⋯
L(s)  = 1  + 0.577·3-s + 2.23·5-s − 1.88·7-s + 2/3·9-s + 1.80·11-s + 1.29·15-s − 1.45·17-s + 0.688·19-s − 1.09·21-s − 5.00·23-s + 2·25-s − 0.192·27-s + 2.69·31-s + 1.04·33-s − 4.22·35-s − 0.164·37-s + 1.24·41-s − 3.96·43-s + 1.49·45-s − 2.04·47-s + 6/7·49-s − 0.840·51-s + 3.29·53-s + 4.04·55-s + 0.397·57-s + 5.37·61-s − 1.25·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{10} \cdot 5^{10} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{10} \cdot 5^{10} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{20} \cdot 3^{10} \cdot 5^{10} \cdot 7^{10}\)
Sign: $1$
Analytic conductor: \(179996.\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{420} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{20} \cdot 3^{10} \cdot 5^{10} \cdot 7^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.440511013\)
\(L(\frac12)\) \(\approx\) \(2.440511013\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T - T^{2} + 4 T^{3} + T^{4} - 7 p T^{5} + p T^{6} + 4 p^{2} T^{7} - p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
5 \( ( 1 - T + T^{2} )^{5} \)
7 \( 1 + 5 T + 19 T^{2} + 6 T^{3} - 13 p T^{4} - 533 T^{5} - 13 p^{2} T^{6} + 6 p^{2} T^{7} + 19 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
good11 \( 1 - 6 T + 51 T^{2} - 234 T^{3} + 1195 T^{4} - 3666 T^{5} + 12622 T^{6} - 19434 T^{7} + 25645 T^{8} + 184704 T^{9} - 509783 T^{10} + 184704 p T^{11} + 25645 p^{2} T^{12} - 19434 p^{3} T^{13} + 12622 p^{4} T^{14} - 3666 p^{5} T^{15} + 1195 p^{6} T^{16} - 234 p^{7} T^{17} + 51 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \)
13 \( 1 - 49 T^{2} + 1364 T^{4} - 28371 T^{6} + 474775 T^{8} - 6687888 T^{10} + 474775 p^{2} T^{12} - 28371 p^{4} T^{14} + 1364 p^{6} T^{16} - 49 p^{8} T^{18} + p^{10} T^{20} \)
17 \( 1 + 6 T - 15 T^{2} - 138 T^{3} + 193 T^{4} + 162 p T^{5} + 4928 T^{6} - 30306 T^{7} - 220151 T^{8} + 325680 T^{9} + 6075881 T^{10} + 325680 p T^{11} - 220151 p^{2} T^{12} - 30306 p^{3} T^{13} + 4928 p^{4} T^{14} + 162 p^{6} T^{15} + 193 p^{6} T^{16} - 138 p^{7} T^{17} - 15 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \)
19 \( 1 - 3 T + 83 T^{2} - 240 T^{3} + 200 p T^{4} - 10380 T^{5} + 124845 T^{6} - 315315 T^{7} + 3213415 T^{8} - 7312140 T^{9} + 67225536 T^{10} - 7312140 p T^{11} + 3213415 p^{2} T^{12} - 315315 p^{3} T^{13} + 124845 p^{4} T^{14} - 10380 p^{5} T^{15} + 200 p^{7} T^{16} - 240 p^{7} T^{17} + 83 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \)
23 \( 1 + 24 T + 338 T^{2} + 3504 T^{3} + 29201 T^{4} + 203310 T^{5} + 1216412 T^{6} + 6430392 T^{7} + 31045441 T^{8} + 143820006 T^{9} + 675418266 T^{10} + 143820006 p T^{11} + 31045441 p^{2} T^{12} + 6430392 p^{3} T^{13} + 1216412 p^{4} T^{14} + 203310 p^{5} T^{15} + 29201 p^{6} T^{16} + 3504 p^{7} T^{17} + 338 p^{8} T^{18} + 24 p^{9} T^{19} + p^{10} T^{20} \)
29 \( 1 - 148 T^{2} + 11438 T^{4} - 619594 T^{6} + 25499257 T^{8} - 826380936 T^{10} + 25499257 p^{2} T^{12} - 619594 p^{4} T^{14} + 11438 p^{6} T^{16} - 148 p^{8} T^{18} + p^{10} T^{20} \)
31 \( 1 - 15 T + 167 T^{2} - 1380 T^{3} + 9668 T^{4} - 61404 T^{5} + 363093 T^{6} - 2129475 T^{7} + 12161347 T^{8} - 68552376 T^{9} + 387742776 T^{10} - 68552376 p T^{11} + 12161347 p^{2} T^{12} - 2129475 p^{3} T^{13} + 363093 p^{4} T^{14} - 61404 p^{5} T^{15} + 9668 p^{6} T^{16} - 1380 p^{7} T^{17} + 167 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \)
37 \( 1 + T - 117 T^{2} - 176 T^{3} + 6668 T^{4} + 11380 T^{5} - 299439 T^{6} - 438173 T^{7} + 12712375 T^{8} + 7307172 T^{9} - 492022188 T^{10} + 7307172 p T^{11} + 12712375 p^{2} T^{12} - 438173 p^{3} T^{13} - 299439 p^{4} T^{14} + 11380 p^{5} T^{15} + 6668 p^{6} T^{16} - 176 p^{7} T^{17} - 117 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \)
41 \( ( 1 - 4 T + 90 T^{2} - 592 T^{3} + 3749 T^{4} - 36434 T^{5} + 3749 p T^{6} - 592 p^{2} T^{7} + 90 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
43 \( ( 1 + 13 T + 173 T^{2} + 1668 T^{3} + 14545 T^{4} + 96933 T^{5} + 14545 p T^{6} + 1668 p^{2} T^{7} + 173 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
47 \( 1 + 14 T + 49 T^{2} - 70 T^{3} - 1449 T^{4} - 16780 T^{5} - 75362 T^{6} + 133028 T^{7} + 1885261 T^{8} + 57961722 T^{9} + 732803503 T^{10} + 57961722 p T^{11} + 1885261 p^{2} T^{12} + 133028 p^{3} T^{13} - 75362 p^{4} T^{14} - 16780 p^{5} T^{15} - 1449 p^{6} T^{16} - 70 p^{7} T^{17} + 49 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \)
53 \( 1 - 24 T + 337 T^{2} - 3480 T^{3} + 28655 T^{4} - 224976 T^{5} + 1724958 T^{6} - 14318448 T^{7} + 120990445 T^{8} - 964554888 T^{9} + 7393360287 T^{10} - 964554888 p T^{11} + 120990445 p^{2} T^{12} - 14318448 p^{3} T^{13} + 1724958 p^{4} T^{14} - 224976 p^{5} T^{15} + 28655 p^{6} T^{16} - 3480 p^{7} T^{17} + 337 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \)
59 \( 1 - 135 T^{2} + 756 T^{3} + 7351 T^{4} - 84450 T^{5} - 41974 T^{6} + 3597210 T^{7} - 8726459 T^{8} - 56020614 T^{9} + 414268763 T^{10} - 56020614 p T^{11} - 8726459 p^{2} T^{12} + 3597210 p^{3} T^{13} - 41974 p^{4} T^{14} - 84450 p^{5} T^{15} + 7351 p^{6} T^{16} + 756 p^{7} T^{17} - 135 p^{8} T^{18} + p^{10} T^{20} \)
61 \( 1 - 42 T + 1112 T^{2} - 22008 T^{3} + 361073 T^{4} - 5062200 T^{5} + 62481246 T^{6} - 687242394 T^{7} + 6821292673 T^{8} - 61325603712 T^{9} + 502052393310 T^{10} - 61325603712 p T^{11} + 6821292673 p^{2} T^{12} - 687242394 p^{3} T^{13} + 62481246 p^{4} T^{14} - 5062200 p^{5} T^{15} + 361073 p^{6} T^{16} - 22008 p^{7} T^{17} + 1112 p^{8} T^{18} - 42 p^{9} T^{19} + p^{10} T^{20} \)
67 \( 1 - 7 T - 94 T^{2} + 1731 T^{3} + 26 T^{4} - 170241 T^{5} + 879508 T^{6} + 10247793 T^{7} - 116682655 T^{8} - 286659444 T^{9} + 9801469996 T^{10} - 286659444 p T^{11} - 116682655 p^{2} T^{12} + 10247793 p^{3} T^{13} + 879508 p^{4} T^{14} - 170241 p^{5} T^{15} + 26 p^{6} T^{16} + 1731 p^{7} T^{17} - 94 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \)
71 \( 1 - 422 T^{2} + 92093 T^{4} - 13410012 T^{6} + 1429352290 T^{8} - 115763220252 T^{10} + 1429352290 p^{2} T^{12} - 13410012 p^{4} T^{14} + 92093 p^{6} T^{16} - 422 p^{8} T^{18} + p^{10} T^{20} \)
73 \( 1 + 3 T + 115 T^{2} + 336 T^{3} + 8912 T^{4} + 58260 T^{5} + 290497 T^{6} + 1508853 T^{7} - 7279937 T^{8} - 55999116 T^{9} - 744209196 T^{10} - 55999116 p T^{11} - 7279937 p^{2} T^{12} + 1508853 p^{3} T^{13} + 290497 p^{4} T^{14} + 58260 p^{5} T^{15} + 8912 p^{6} T^{16} + 336 p^{7} T^{17} + 115 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \)
79 \( 1 - T - 201 T^{2} + 740 T^{3} + 18128 T^{4} - 100900 T^{5} - 805347 T^{6} + 5115611 T^{7} + 21370651 T^{8} - 77003688 T^{9} - 699661200 T^{10} - 77003688 p T^{11} + 21370651 p^{2} T^{12} + 5115611 p^{3} T^{13} - 805347 p^{4} T^{14} - 100900 p^{5} T^{15} + 18128 p^{6} T^{16} + 740 p^{7} T^{17} - 201 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \)
83 \( ( 1 - 4 T + 36 T^{2} - 304 T^{3} + 3029 T^{4} + 33442 T^{5} + 3029 p T^{6} - 304 p^{2} T^{7} + 36 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
89 \( 1 + 28 T + 118 T^{2} - 2240 T^{3} + 12999 T^{4} + 7142 p T^{5} + 2053984 T^{6} - 15180164 T^{7} + 276922069 T^{8} + 3387791550 T^{9} + 10662822610 T^{10} + 3387791550 p T^{11} + 276922069 p^{2} T^{12} - 15180164 p^{3} T^{13} + 2053984 p^{4} T^{14} + 7142 p^{6} T^{15} + 12999 p^{6} T^{16} - 2240 p^{7} T^{17} + 118 p^{8} T^{18} + 28 p^{9} T^{19} + p^{10} T^{20} \)
97 \( 1 - 758 T^{2} + 269933 T^{4} - 59888792 T^{6} + 9235543570 T^{8} - 1040392525668 T^{10} + 9235543570 p^{2} T^{12} - 59888792 p^{4} T^{14} + 269933 p^{6} T^{16} - 758 p^{8} T^{18} + p^{10} T^{20} \)
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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

−4.14408008463203194631867713191, −4.12469961719517472602044917808, −4.02911342424338911496998107511, −3.92241844442300222375718619344, −3.86127421000909461890691022396, −3.72814948360219399748619766156, −3.44581260108048289068143846719, −3.32108269682448875489956636855, −3.24564123595883957356765862671, −3.17668275280752308422342574141, −2.95967889714600831243729318901, −2.75513251753761082689450940639, −2.61687245340190922518673525995, −2.50205350593252423794052630243, −2.45053773858738151446264261704, −2.08499251404113868543980640584, −2.03923454789571647572753713555, −1.97842063142399830043493053692, −1.93016579008400675698332641412, −1.68501250288476466254959133600, −1.38338971620188910431068467300, −1.24413183542557531365773490939, −1.08997491998934139769388866792, −0.61647495490232449492829823577, −0.22000748975483652772892720898, 0.22000748975483652772892720898, 0.61647495490232449492829823577, 1.08997491998934139769388866792, 1.24413183542557531365773490939, 1.38338971620188910431068467300, 1.68501250288476466254959133600, 1.93016579008400675698332641412, 1.97842063142399830043493053692, 2.03923454789571647572753713555, 2.08499251404113868543980640584, 2.45053773858738151446264261704, 2.50205350593252423794052630243, 2.61687245340190922518673525995, 2.75513251753761082689450940639, 2.95967889714600831243729318901, 3.17668275280752308422342574141, 3.24564123595883957356765862671, 3.32108269682448875489956636855, 3.44581260108048289068143846719, 3.72814948360219399748619766156, 3.86127421000909461890691022396, 3.92241844442300222375718619344, 4.02911342424338911496998107511, 4.12469961719517472602044917808, 4.14408008463203194631867713191

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

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