Properties

Degree 20
Conductor $ 2^{10} \cdot 5^{10} \cdot 401^{10} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 10

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·2-s − 4·3-s + 55·4-s + 10·5-s + 40·6-s − 3·7-s − 220·8-s − 4·9-s − 100·10-s − 11·11-s − 220·12-s + 6·13-s + 30·14-s − 40·15-s + 715·16-s + 9·17-s + 40·18-s − 13·19-s + 550·20-s + 12·21-s + 110·22-s − 3·23-s + 880·24-s + 55·25-s − 60·26-s + 38·27-s − 165·28-s + ⋯
L(s)  = 1  − 7.07·2-s − 2.30·3-s + 55/2·4-s + 4.47·5-s + 16.3·6-s − 1.13·7-s − 77.7·8-s − 4/3·9-s − 31.6·10-s − 3.31·11-s − 63.5·12-s + 1.66·13-s + 8.01·14-s − 10.3·15-s + 178.·16-s + 2.18·17-s + 9.42·18-s − 2.98·19-s + 122.·20-s + 2.61·21-s + 23.4·22-s − 0.625·23-s + 179.·24-s + 11·25-s − 11.7·26-s + 7.31·27-s − 31.1·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 5^{10} \cdot 401^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 5^{10} \cdot 401^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(20\)
\( N \)  =  \(2^{10} \cdot 5^{10} \cdot 401^{10}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4010} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  10
Selberg data  =  $(20,\ 2^{10} \cdot 5^{10} \cdot 401^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;401\}$,\(F_p(T)\) is a polynomial of degree 20. If $p \in \{2,\;5,\;401\}$, then $F_p(T)$ is a polynomial of degree at most 19.
$p$$F_p(T)$
bad2 \( ( 1 + T )^{10} \)
5 \( ( 1 - T )^{10} \)
401 \( ( 1 - T )^{10} \)
good3 \( 1 + 4 T + 20 T^{2} + 58 T^{3} + 182 T^{4} + 434 T^{5} + 1076 T^{6} + 745 p T^{7} + 4753 T^{8} + 8719 T^{9} + 16208 T^{10} + 8719 p T^{11} + 4753 p^{2} T^{12} + 745 p^{4} T^{13} + 1076 p^{4} T^{14} + 434 p^{5} T^{15} + 182 p^{6} T^{16} + 58 p^{7} T^{17} + 20 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \)
7 \( 1 + 3 T + 6 p T^{2} + 129 T^{3} + 894 T^{4} + 2832 T^{5} + 12609 T^{6} + 5675 p T^{7} + 2657 p^{2} T^{8} + 387735 T^{9} + 147578 p T^{10} + 387735 p T^{11} + 2657 p^{4} T^{12} + 5675 p^{4} T^{13} + 12609 p^{4} T^{14} + 2832 p^{5} T^{15} + 894 p^{6} T^{16} + 129 p^{7} T^{17} + 6 p^{9} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \)
11 \( 1 + p T + 103 T^{2} + 56 p T^{3} + 3268 T^{4} + 13096 T^{5} + 4351 p T^{6} + 133699 T^{7} + 32817 p T^{8} + 744096 T^{9} + 213640 p T^{10} + 744096 p T^{11} + 32817 p^{3} T^{12} + 133699 p^{3} T^{13} + 4351 p^{5} T^{14} + 13096 p^{5} T^{15} + 3268 p^{6} T^{16} + 56 p^{8} T^{17} + 103 p^{8} T^{18} + p^{10} T^{19} + p^{10} T^{20} \)
13 \( 1 - 6 T + 81 T^{2} - 287 T^{3} + 2372 T^{4} - 4060 T^{5} + 34023 T^{6} + 16767 T^{7} + 283911 T^{8} + 1210928 T^{9} + 2420282 T^{10} + 1210928 p T^{11} + 283911 p^{2} T^{12} + 16767 p^{3} T^{13} + 34023 p^{4} T^{14} - 4060 p^{5} T^{15} + 2372 p^{6} T^{16} - 287 p^{7} T^{17} + 81 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \)
17 \( 1 - 9 T + 152 T^{2} - 61 p T^{3} + 10141 T^{4} - 56380 T^{5} + 409608 T^{6} - 1923285 T^{7} + 11325598 T^{8} - 45513113 T^{9} + 225415358 T^{10} - 45513113 p T^{11} + 11325598 p^{2} T^{12} - 1923285 p^{3} T^{13} + 409608 p^{4} T^{14} - 56380 p^{5} T^{15} + 10141 p^{6} T^{16} - 61 p^{8} T^{17} + 152 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \)
19 \( 1 + 13 T + 192 T^{2} + 1664 T^{3} + 14659 T^{4} + 97713 T^{5} + 648691 T^{6} + 3562244 T^{7} + 19446002 T^{8} + 91270928 T^{9} + 426588208 T^{10} + 91270928 p T^{11} + 19446002 p^{2} T^{12} + 3562244 p^{3} T^{13} + 648691 p^{4} T^{14} + 97713 p^{5} T^{15} + 14659 p^{6} T^{16} + 1664 p^{7} T^{17} + 192 p^{8} T^{18} + 13 p^{9} T^{19} + p^{10} T^{20} \)
23 \( 1 + 3 T + 187 T^{2} + 425 T^{3} + 16093 T^{4} + 27993 T^{5} + 855348 T^{6} + 1169460 T^{7} + 31480234 T^{8} + 35291365 T^{9} + 842951618 T^{10} + 35291365 p T^{11} + 31480234 p^{2} T^{12} + 1169460 p^{3} T^{13} + 855348 p^{4} T^{14} + 27993 p^{5} T^{15} + 16093 p^{6} T^{16} + 425 p^{7} T^{17} + 187 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \)
29 \( 1 + 4 T + 129 T^{2} + 570 T^{3} + 7726 T^{4} + 35616 T^{5} + 303517 T^{6} + 1267961 T^{7} + 9743651 T^{8} + 32792993 T^{9} + 289804684 T^{10} + 32792993 p T^{11} + 9743651 p^{2} T^{12} + 1267961 p^{3} T^{13} + 303517 p^{4} T^{14} + 35616 p^{5} T^{15} + 7726 p^{6} T^{16} + 570 p^{7} T^{17} + 129 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \)
31 \( 1 + 17 T + 216 T^{2} + 2088 T^{3} + 18749 T^{4} + 145342 T^{5} + 1063788 T^{6} + 7150461 T^{7} + 46404448 T^{8} + 277307962 T^{9} + 1596171690 T^{10} + 277307962 p T^{11} + 46404448 p^{2} T^{12} + 7150461 p^{3} T^{13} + 1063788 p^{4} T^{14} + 145342 p^{5} T^{15} + 18749 p^{6} T^{16} + 2088 p^{7} T^{17} + 216 p^{8} T^{18} + 17 p^{9} T^{19} + p^{10} T^{20} \)
37 \( 1 + 15 T + 318 T^{2} + 3739 T^{3} + 45945 T^{4} + 441510 T^{5} + 4052846 T^{6} + 32751775 T^{7} + 244564024 T^{8} + 1686062901 T^{9} + 10602280982 T^{10} + 1686062901 p T^{11} + 244564024 p^{2} T^{12} + 32751775 p^{3} T^{13} + 4052846 p^{4} T^{14} + 441510 p^{5} T^{15} + 45945 p^{6} T^{16} + 3739 p^{7} T^{17} + 318 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \)
41 \( 1 + 11 T + 238 T^{2} + 2000 T^{3} + 28011 T^{4} + 207360 T^{5} + 2280094 T^{6} + 14946485 T^{7} + 137235898 T^{8} + 799593928 T^{9} + 6373618252 T^{10} + 799593928 p T^{11} + 137235898 p^{2} T^{12} + 14946485 p^{3} T^{13} + 2280094 p^{4} T^{14} + 207360 p^{5} T^{15} + 28011 p^{6} T^{16} + 2000 p^{7} T^{17} + 238 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \)
43 \( 1 + 11 T + 197 T^{2} + 1901 T^{3} + 22094 T^{4} + 191017 T^{5} + 1740213 T^{6} + 13397158 T^{7} + 106178861 T^{8} + 725466695 T^{9} + 5107982132 T^{10} + 725466695 p T^{11} + 106178861 p^{2} T^{12} + 13397158 p^{3} T^{13} + 1740213 p^{4} T^{14} + 191017 p^{5} T^{15} + 22094 p^{6} T^{16} + 1901 p^{7} T^{17} + 197 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \)
47 \( 1 - 3 T + 194 T^{2} - 323 T^{3} + 17109 T^{4} - 20 p T^{5} + 811722 T^{6} + 2405797 T^{7} + 19065686 T^{8} + 231609061 T^{9} + 317343936 T^{10} + 231609061 p T^{11} + 19065686 p^{2} T^{12} + 2405797 p^{3} T^{13} + 811722 p^{4} T^{14} - 20 p^{6} T^{15} + 17109 p^{6} T^{16} - 323 p^{7} T^{17} + 194 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \)
53 \( 1 - 25 T + 427 T^{2} - 5582 T^{3} + 64602 T^{4} - 665382 T^{5} + 6336438 T^{6} - 56080783 T^{7} + 467737103 T^{8} - 3708855556 T^{9} + 27786225496 T^{10} - 3708855556 p T^{11} + 467737103 p^{2} T^{12} - 56080783 p^{3} T^{13} + 6336438 p^{4} T^{14} - 665382 p^{5} T^{15} + 64602 p^{6} T^{16} - 5582 p^{7} T^{17} + 427 p^{8} T^{18} - 25 p^{9} T^{19} + p^{10} T^{20} \)
59 \( 1 + 46 T + 1370 T^{2} + 29248 T^{3} + 507730 T^{4} + 7358451 T^{5} + 92806871 T^{6} + 1028827578 T^{7} + 10218323913 T^{8} + 91125712475 T^{9} + 736477435704 T^{10} + 91125712475 p T^{11} + 10218323913 p^{2} T^{12} + 1028827578 p^{3} T^{13} + 92806871 p^{4} T^{14} + 7358451 p^{5} T^{15} + 507730 p^{6} T^{16} + 29248 p^{7} T^{17} + 1370 p^{8} T^{18} + 46 p^{9} T^{19} + p^{10} T^{20} \)
61 \( 1 + 54 T + 1716 T^{2} + 39282 T^{3} + 715319 T^{4} + 10864488 T^{5} + 141988979 T^{6} + 1627514861 T^{7} + 16582577610 T^{8} + 151415379327 T^{9} + 1245270340302 T^{10} + 151415379327 p T^{11} + 16582577610 p^{2} T^{12} + 1627514861 p^{3} T^{13} + 141988979 p^{4} T^{14} + 10864488 p^{5} T^{15} + 715319 p^{6} T^{16} + 39282 p^{7} T^{17} + 1716 p^{8} T^{18} + 54 p^{9} T^{19} + p^{10} T^{20} \)
67 \( 1 + 26 T + 703 T^{2} + 12197 T^{3} + 199474 T^{4} + 2642918 T^{5} + 32795787 T^{6} + 354040609 T^{7} + 3605991281 T^{8} + 32798357058 T^{9} + 283115362804 T^{10} + 32798357058 p T^{11} + 3605991281 p^{2} T^{12} + 354040609 p^{3} T^{13} + 32795787 p^{4} T^{14} + 2642918 p^{5} T^{15} + 199474 p^{6} T^{16} + 12197 p^{7} T^{17} + 703 p^{8} T^{18} + 26 p^{9} T^{19} + p^{10} T^{20} \)
71 \( 1 + 16 T + 658 T^{2} + 7807 T^{3} + 181148 T^{4} + 1676154 T^{5} + 28845062 T^{6} + 216422887 T^{7} + 3107133973 T^{8} + 19751330864 T^{9} + 250233628330 T^{10} + 19751330864 p T^{11} + 3107133973 p^{2} T^{12} + 216422887 p^{3} T^{13} + 28845062 p^{4} T^{14} + 1676154 p^{5} T^{15} + 181148 p^{6} T^{16} + 7807 p^{7} T^{17} + 658 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \)
73 \( 1 - 4 T + 516 T^{2} - 1626 T^{3} + 129718 T^{4} - 334152 T^{5} + 20927890 T^{6} - 45246887 T^{7} + 2391701595 T^{8} - 4417520027 T^{9} + 201898642960 T^{10} - 4417520027 p T^{11} + 2391701595 p^{2} T^{12} - 45246887 p^{3} T^{13} + 20927890 p^{4} T^{14} - 334152 p^{5} T^{15} + 129718 p^{6} T^{16} - 1626 p^{7} T^{17} + 516 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \)
79 \( 1 + 19 T + 701 T^{2} + 10669 T^{3} + 223488 T^{4} + 2834077 T^{5} + 43314051 T^{6} + 466164732 T^{7} + 5686462673 T^{8} + 52208581403 T^{9} + 529742938534 T^{10} + 52208581403 p T^{11} + 5686462673 p^{2} T^{12} + 466164732 p^{3} T^{13} + 43314051 p^{4} T^{14} + 2834077 p^{5} T^{15} + 223488 p^{6} T^{16} + 10669 p^{7} T^{17} + 701 p^{8} T^{18} + 19 p^{9} T^{19} + p^{10} T^{20} \)
83 \( 1 - 19 T + 687 T^{2} - 11547 T^{3} + 227142 T^{4} - 3257653 T^{5} + 46742113 T^{6} - 563300086 T^{7} + 6536553969 T^{8} - 66156590137 T^{9} + 643472849516 T^{10} - 66156590137 p T^{11} + 6536553969 p^{2} T^{12} - 563300086 p^{3} T^{13} + 46742113 p^{4} T^{14} - 3257653 p^{5} T^{15} + 227142 p^{6} T^{16} - 11547 p^{7} T^{17} + 687 p^{8} T^{18} - 19 p^{9} T^{19} + p^{10} T^{20} \)
89 \( 1 + 30 T + 1011 T^{2} + 20699 T^{3} + 418614 T^{4} + 6544860 T^{5} + 98981835 T^{6} + 1250450097 T^{7} + 15230285299 T^{8} + 159866982622 T^{9} + 1617442689892 T^{10} + 159866982622 p T^{11} + 15230285299 p^{2} T^{12} + 1250450097 p^{3} T^{13} + 98981835 p^{4} T^{14} + 6544860 p^{5} T^{15} + 418614 p^{6} T^{16} + 20699 p^{7} T^{17} + 1011 p^{8} T^{18} + 30 p^{9} T^{19} + p^{10} T^{20} \)
97 \( 1 + 16 T + 640 T^{2} + 6946 T^{3} + 179074 T^{4} + 1534105 T^{5} + 33304185 T^{6} + 244454186 T^{7} + 4690150175 T^{8} + 30216640161 T^{9} + 513900985612 T^{10} + 30216640161 p T^{11} + 4690150175 p^{2} T^{12} + 244454186 p^{3} T^{13} + 33304185 p^{4} T^{14} + 1534105 p^{5} T^{15} + 179074 p^{6} T^{16} + 6946 p^{7} T^{17} + 640 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.24724735339100871088376678597, −3.16206429490263436149127001606, −2.99929515648955644351436086930, −2.95551592826889692691714255862, −2.87668056435959380789085571934, −2.69341027358928163045947533213, −2.55975168377336365124557300766, −2.49561102606008902058867743273, −2.47730620088101945932137289346, −2.39532974877939007520138977870, −2.31396593164301114945011574558, −2.29497988778835994437268230440, −2.21741838233280892897559126398, −1.95860501700622797059490697609, −1.91963133058607659587934320137, −1.63766370577317880636364761357, −1.56026130152732342930161177109, −1.52385849618252771133148391166, −1.42440661892480559643153495973, −1.39853409528883700262324910648, −1.38714389737662786671511790870, −1.22402491260016228744459577363, −1.18638041785933017448880353192, −1.14497977190670319467270237010, −1.02625507922840265802455436450, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.02625507922840265802455436450, 1.14497977190670319467270237010, 1.18638041785933017448880353192, 1.22402491260016228744459577363, 1.38714389737662786671511790870, 1.39853409528883700262324910648, 1.42440661892480559643153495973, 1.52385849618252771133148391166, 1.56026130152732342930161177109, 1.63766370577317880636364761357, 1.91963133058607659587934320137, 1.95860501700622797059490697609, 2.21741838233280892897559126398, 2.29497988778835994437268230440, 2.31396593164301114945011574558, 2.39532974877939007520138977870, 2.47730620088101945932137289346, 2.49561102606008902058867743273, 2.55975168377336365124557300766, 2.69341027358928163045947533213, 2.87668056435959380789085571934, 2.95551592826889692691714255862, 2.99929515648955644351436086930, 3.16206429490263436149127001606, 3.24724735339100871088376678597

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

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