Properties

Label 20-1323e10-1.1-c1e10-0-4
Degree $20$
Conductor $1.643\times 10^{31}$
Sign $1$
Analytic cond. $1.73128\times 10^{10}$
Root an. cond. $3.25026$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 5·4-s − 8·5-s − 8·8-s + 16·10-s + 8·11-s + 8·13-s + 15·16-s + 12·17-s − 19-s − 40·20-s − 16·22-s + 6·23-s + 8·25-s − 16·26-s − 7·29-s + 3·31-s − 20·32-s − 24·34-s + 2·38-s + 64·40-s + 5·41-s − 7·43-s + 40·44-s − 12·46-s + 27·47-s − 16·50-s + ⋯
L(s)  = 1  − 1.41·2-s + 5/2·4-s − 3.57·5-s − 2.82·8-s + 5.05·10-s + 2.41·11-s + 2.21·13-s + 15/4·16-s + 2.91·17-s − 0.229·19-s − 8.94·20-s − 3.41·22-s + 1.25·23-s + 8/5·25-s − 3.13·26-s − 1.29·29-s + 0.538·31-s − 3.53·32-s − 4.11·34-s + 0.324·38-s + 10.1·40-s + 0.780·41-s − 1.06·43-s + 6.03·44-s − 1.76·46-s + 3.93·47-s − 2.26·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{30} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{30} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(3^{30} \cdot 7^{20}\)
Sign: $1$
Analytic conductor: \(1.73128\times 10^{10}\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1323} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 3^{30} \cdot 7^{20} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.763558642\)
\(L(\frac12)\) \(\approx\) \(1.763558642\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + p T - T^{2} - p^{2} T^{3} - p T^{4} - p T^{5} + 3 p T^{6} + 21 T^{7} + 3 p T^{8} - 13 T^{9} - 5 T^{10} - 13 p T^{11} + 3 p^{3} T^{12} + 21 p^{3} T^{13} + 3 p^{5} T^{14} - p^{6} T^{15} - p^{7} T^{16} - p^{9} T^{17} - p^{8} T^{18} + p^{10} T^{19} + p^{10} T^{20} \)
5 \( ( 1 + 4 T + 4 p T^{2} + 62 T^{3} + 193 T^{4} + 423 T^{5} + 193 p T^{6} + 62 p^{2} T^{7} + 4 p^{4} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
11 \( ( 1 - 4 T + 47 T^{2} - 161 T^{3} + 958 T^{4} - 2589 T^{5} + 958 p T^{6} - 161 p^{2} T^{7} + 47 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
13 \( 1 - 8 T - 14 T^{2} + 14 p T^{3} + 686 T^{4} - 4429 T^{5} - 12871 T^{6} + 3323 p T^{7} + 305249 T^{8} - 358672 T^{9} - 3841969 T^{10} - 358672 p T^{11} + 305249 p^{2} T^{12} + 3323 p^{4} T^{13} - 12871 p^{4} T^{14} - 4429 p^{5} T^{15} + 686 p^{6} T^{16} + 14 p^{8} T^{17} - 14 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \)
17 \( 1 - 12 T + 14 T^{2} + 192 T^{3} + 1185 T^{4} - 11847 T^{5} - 6180 T^{6} + 65736 T^{7} + 1002861 T^{8} - 2436261 T^{9} - 7749777 T^{10} - 2436261 p T^{11} + 1002861 p^{2} T^{12} + 65736 p^{3} T^{13} - 6180 p^{4} T^{14} - 11847 p^{5} T^{15} + 1185 p^{6} T^{16} + 192 p^{7} T^{17} + 14 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \)
19 \( 1 + T - 53 T^{2} - 10 p T^{3} + 1262 T^{4} + 7007 T^{5} - 13111 T^{6} - 116110 T^{7} + 67964 T^{8} + 721616 T^{9} - 440023 T^{10} + 721616 p T^{11} + 67964 p^{2} T^{12} - 116110 p^{3} T^{13} - 13111 p^{4} T^{14} + 7007 p^{5} T^{15} + 1262 p^{6} T^{16} - 10 p^{8} T^{17} - 53 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \)
23 \( ( 1 - 3 T + 52 T^{2} - 225 T^{3} + 2023 T^{4} - 5565 T^{5} + 2023 p T^{6} - 225 p^{2} T^{7} + 52 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
29 \( 1 + 7 T - 76 T^{2} - 419 T^{3} + 4561 T^{4} + 15146 T^{5} - 199563 T^{6} - 341373 T^{7} + 6918636 T^{8} + 2570041 T^{9} - 219913241 T^{10} + 2570041 p T^{11} + 6918636 p^{2} T^{12} - 341373 p^{3} T^{13} - 199563 p^{4} T^{14} + 15146 p^{5} T^{15} + 4561 p^{6} T^{16} - 419 p^{7} T^{17} - 76 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \)
31 \( 1 - 3 T - 125 T^{2} + 214 T^{3} + 9282 T^{4} - 8387 T^{5} - 503981 T^{6} + 245082 T^{7} + 21459514 T^{8} - 116758 p T^{9} - 734820027 T^{10} - 116758 p^{2} T^{11} + 21459514 p^{2} T^{12} + 245082 p^{3} T^{13} - 503981 p^{4} T^{14} - 8387 p^{5} T^{15} + 9282 p^{6} T^{16} + 214 p^{7} T^{17} - 125 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \)
37 \( 1 - 89 T^{2} + 560 T^{3} + 4503 T^{4} - 45352 T^{5} + 27130 T^{6} + 2296536 T^{7} - 9801827 T^{8} - 33131096 T^{9} + 610977105 T^{10} - 33131096 p T^{11} - 9801827 p^{2} T^{12} + 2296536 p^{3} T^{13} + 27130 p^{4} T^{14} - 45352 p^{5} T^{15} + 4503 p^{6} T^{16} + 560 p^{7} T^{17} - 89 p^{8} T^{18} + p^{10} T^{20} \)
41 \( 1 - 5 T - 136 T^{2} + 733 T^{3} + 10507 T^{4} - 54412 T^{5} - 554055 T^{6} + 2345451 T^{7} + 23706084 T^{8} - 41392439 T^{9} - 952045937 T^{10} - 41392439 p T^{11} + 23706084 p^{2} T^{12} + 2345451 p^{3} T^{13} - 554055 p^{4} T^{14} - 54412 p^{5} T^{15} + 10507 p^{6} T^{16} + 733 p^{7} T^{17} - 136 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \)
43 \( 1 + 7 T - 77 T^{2} - 66 T^{3} + 7014 T^{4} - 3843 T^{5} - 95427 T^{6} + 1632678 T^{7} - 3708600 T^{8} - 15416324 T^{9} + 670279801 T^{10} - 15416324 p T^{11} - 3708600 p^{2} T^{12} + 1632678 p^{3} T^{13} - 95427 p^{4} T^{14} - 3843 p^{5} T^{15} + 7014 p^{6} T^{16} - 66 p^{7} T^{17} - 77 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \)
47 \( 1 - 27 T + 281 T^{2} - 1758 T^{3} + 13050 T^{4} - 78783 T^{5} - 25248 T^{6} + 1518381 T^{7} + 9454350 T^{8} - 53043051 T^{9} - 242331903 T^{10} - 53043051 p T^{11} + 9454350 p^{2} T^{12} + 1518381 p^{3} T^{13} - 25248 p^{4} T^{14} - 78783 p^{5} T^{15} + 13050 p^{6} T^{16} - 1758 p^{7} T^{17} + 281 p^{8} T^{18} - 27 p^{9} T^{19} + p^{10} T^{20} \)
53 \( 1 - 21 T + 41 T^{2} + 924 T^{3} + 12966 T^{4} - 177027 T^{5} - 601755 T^{6} + 3783942 T^{7} + 110973258 T^{8} - 340111866 T^{9} - 4044436041 T^{10} - 340111866 p T^{11} + 110973258 p^{2} T^{12} + 3783942 p^{3} T^{13} - 601755 p^{4} T^{14} - 177027 p^{5} T^{15} + 12966 p^{6} T^{16} + 924 p^{7} T^{17} + 41 p^{8} T^{18} - 21 p^{9} T^{19} + p^{10} T^{20} \)
59 \( 1 - 30 T + 299 T^{2} - 1644 T^{3} + 26547 T^{4} - 5838 p T^{5} + 1635267 T^{6} - 9620487 T^{7} + 170035344 T^{8} - 1056366303 T^{9} + 3109579647 T^{10} - 1056366303 p T^{11} + 170035344 p^{2} T^{12} - 9620487 p^{3} T^{13} + 1635267 p^{4} T^{14} - 5838 p^{6} T^{15} + 26547 p^{6} T^{16} - 1644 p^{7} T^{17} + 299 p^{8} T^{18} - 30 p^{9} T^{19} + p^{10} T^{20} \)
61 \( 1 - 14 T - 143 T^{2} + 2072 T^{3} + 23777 T^{4} - 251656 T^{5} - 2164351 T^{6} + 13562879 T^{7} + 202896254 T^{8} - 466067647 T^{9} - 12461386219 T^{10} - 466067647 p T^{11} + 202896254 p^{2} T^{12} + 13562879 p^{3} T^{13} - 2164351 p^{4} T^{14} - 251656 p^{5} T^{15} + 23777 p^{6} T^{16} + 2072 p^{7} T^{17} - 143 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \)
67 \( 1 + 2 T - 128 T^{2} - 128 T^{3} + 6161 T^{4} - 2183 T^{5} + 29300 T^{6} + 394018 T^{7} - 17169907 T^{8} - 2850929 T^{9} + 1197895103 T^{10} - 2850929 p T^{11} - 17169907 p^{2} T^{12} + 394018 p^{3} T^{13} + 29300 p^{4} T^{14} - 2183 p^{5} T^{15} + 6161 p^{6} T^{16} - 128 p^{7} T^{17} - 128 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \)
71 \( ( 1 - 3 T + 187 T^{2} - 285 T^{3} + 15679 T^{4} - 10143 T^{5} + 15679 p T^{6} - 285 p^{2} T^{7} + 187 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
73 \( 1 + 15 T - 134 T^{2} - 2501 T^{3} + 16563 T^{4} + 235276 T^{5} - 2002535 T^{6} - 9021201 T^{7} + 288508378 T^{8} + 238799411 T^{9} - 25271949561 T^{10} + 238799411 p T^{11} + 288508378 p^{2} T^{12} - 9021201 p^{3} T^{13} - 2002535 p^{4} T^{14} + 235276 p^{5} T^{15} + 16563 p^{6} T^{16} - 2501 p^{7} T^{17} - 134 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \)
79 \( 1 + 4 T - 284 T^{2} - 1776 T^{3} + 44175 T^{4} + 312399 T^{5} - 4187754 T^{6} - 29772300 T^{7} + 295992489 T^{8} + 1067553919 T^{9} - 20151634301 T^{10} + 1067553919 p T^{11} + 295992489 p^{2} T^{12} - 29772300 p^{3} T^{13} - 4187754 p^{4} T^{14} + 312399 p^{5} T^{15} + 44175 p^{6} T^{16} - 1776 p^{7} T^{17} - 284 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \)
83 \( 1 - 9 T - 148 T^{2} - 297 T^{3} + 24654 T^{4} + 118125 T^{5} - 807174 T^{6} - 21382137 T^{7} - 37648479 T^{8} + 452536146 T^{9} + 15509586612 T^{10} + 452536146 p T^{11} - 37648479 p^{2} T^{12} - 21382137 p^{3} T^{13} - 807174 p^{4} T^{14} + 118125 p^{5} T^{15} + 24654 p^{6} T^{16} - 297 p^{7} T^{17} - 148 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \)
89 \( 1 - 28 T + 104 T^{2} + 1736 T^{3} + 31273 T^{4} - 611939 T^{5} - 1780638 T^{6} + 18973932 T^{7} + 740914101 T^{8} - 3271180573 T^{9} - 40614588329 T^{10} - 3271180573 p T^{11} + 740914101 p^{2} T^{12} + 18973932 p^{3} T^{13} - 1780638 p^{4} T^{14} - 611939 p^{5} T^{15} + 31273 p^{6} T^{16} + 1736 p^{7} T^{17} + 104 p^{8} T^{18} - 28 p^{9} T^{19} + p^{10} T^{20} \)
97 \( 1 - 12 T - 197 T^{2} + 1534 T^{3} + 27813 T^{4} - 14090 T^{5} - 4545035 T^{6} + 6881349 T^{7} + 472663750 T^{8} - 908843245 T^{9} - 38512186359 T^{10} - 908843245 p T^{11} + 472663750 p^{2} T^{12} + 6881349 p^{3} T^{13} - 4545035 p^{4} T^{14} - 14090 p^{5} T^{15} + 27813 p^{6} T^{16} + 1534 p^{7} T^{17} - 197 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \)
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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

−3.60791196902328252221210596101, −3.56655419505021388298811996358, −3.32811648214157249347660564913, −3.16188406493983021317666180037, −3.00983386344282705677436880061, −2.96003217703926217581328118601, −2.90868165802798821338704216680, −2.68840244572170762736917920629, −2.56409677723269702711564892018, −2.38341613396539103789395400809, −2.28480527407416353473641099712, −2.23788298637031179939629658456, −2.13000516330102814014400462126, −2.06739179871099086311785061630, −1.67877012269408684001893183902, −1.64947092608875500987255650353, −1.39805608919285873414315929541, −1.28173082317204095062584371805, −1.21273580301306149368209862530, −1.07912701134769507157136257747, −1.00448553112002162110624728045, −0.809517273114771430126286713195, −0.74056972453790423242065736501, −0.35361142095204724433957693583, −0.17273109107289023565981315062, 0.17273109107289023565981315062, 0.35361142095204724433957693583, 0.74056972453790423242065736501, 0.809517273114771430126286713195, 1.00448553112002162110624728045, 1.07912701134769507157136257747, 1.21273580301306149368209862530, 1.28173082317204095062584371805, 1.39805608919285873414315929541, 1.64947092608875500987255650353, 1.67877012269408684001893183902, 2.06739179871099086311785061630, 2.13000516330102814014400462126, 2.23788298637031179939629658456, 2.28480527407416353473641099712, 2.38341613396539103789395400809, 2.56409677723269702711564892018, 2.68840244572170762736917920629, 2.90868165802798821338704216680, 2.96003217703926217581328118601, 3.00983386344282705677436880061, 3.16188406493983021317666180037, 3.32811648214157249347660564913, 3.56655419505021388298811996358, 3.60791196902328252221210596101

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

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