Properties

Degree $20$
Conductor $2.782\times 10^{26}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 5·4-s − 4·5-s + 2·6-s + 8·8-s − 3·9-s − 8·10-s + 4·11-s + 5·12-s + 8·13-s − 4·15-s + 15·16-s + 24·17-s − 6·18-s + 2·19-s − 20·20-s + 8·22-s + 3·23-s + 8·24-s + 20·25-s + 16·26-s − 27-s + 7·29-s − 8·30-s + 3·31-s + 20·32-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 5/2·4-s − 1.78·5-s + 0.816·6-s + 2.82·8-s − 9-s − 2.52·10-s + 1.20·11-s + 1.44·12-s + 2.21·13-s − 1.03·15-s + 15/4·16-s + 5.82·17-s − 1.41·18-s + 0.458·19-s − 4.47·20-s + 1.70·22-s + 0.625·23-s + 1.63·24-s + 4·25-s + 3.13·26-s − 0.192·27-s + 1.29·29-s − 1.46·30-s + 0.538·31-s + 3.53·32-s + ⋯

Functional equation

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

Invariants

Degree: \(20\)
Conductor: \(3^{20} \cdot 7^{20}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 3^{20} \cdot 7^{20} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(47.8887\)
\(L(\frac12)\) \(\approx\) \(47.8887\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T + 4 T^{2} - 2 p T^{3} + 2 p^{2} T^{4} - p^{2} T^{5} + 2 p^{3} T^{6} - 2 p^{3} T^{7} + 4 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
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 \( 1 + 4 T - 4 T^{2} - 44 T^{3} - 41 T^{4} + 119 T^{5} + 222 T^{6} + 456 T^{7} + 1623 T^{8} - 2021 T^{9} - 16541 T^{10} - 2021 p T^{11} + 1623 p^{2} T^{12} + 456 p^{3} T^{13} + 222 p^{4} T^{14} + 119 p^{5} T^{15} - 41 p^{6} T^{16} - 44 p^{7} T^{17} - 4 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \)
11 \( 1 - 4 T - 31 T^{2} + 134 T^{3} + 607 T^{4} - 2492 T^{5} - 8385 T^{6} + 27495 T^{7} + 98940 T^{8} - 135733 T^{9} - 1043873 T^{10} - 135733 p T^{11} + 98940 p^{2} T^{12} + 27495 p^{3} T^{13} - 8385 p^{4} T^{14} - 2492 p^{5} T^{15} + 607 p^{6} T^{16} + 134 p^{7} T^{17} - 31 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \)
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 \( ( 1 - 12 T + 130 T^{2} - 876 T^{3} + 5203 T^{4} - 22839 T^{5} + 5203 p T^{6} - 876 p^{2} T^{7} + 130 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
19 \( ( 1 - T + 54 T^{2} - 122 T^{3} + 1532 T^{4} - 3483 T^{5} + 1532 p T^{6} - 122 p^{2} T^{7} + 54 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \)
23 \( 1 - 3 T - 43 T^{2} + 294 T^{3} + 6 T^{4} - 5127 T^{5} + 21792 T^{6} - 135027 T^{7} + 502362 T^{8} + 3271749 T^{9} - 33095343 T^{10} + 3271749 p T^{11} + 502362 p^{2} T^{12} - 135027 p^{3} T^{13} + 21792 p^{4} T^{14} - 5127 p^{5} T^{15} + 6 p^{6} T^{16} + 294 p^{7} T^{17} - 43 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \)
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 \( ( 1 + 89 T^{2} + 280 T^{3} + 3418 T^{4} + 20432 T^{5} + 3418 p T^{6} + 280 p^{2} T^{7} + 89 p^{3} T^{8} + p^{5} T^{10} )^{2} \)
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 \( ( 1 - 21 T + 400 T^{2} - 4662 T^{3} + 49132 T^{4} - 375771 T^{5} + 49132 p T^{6} - 4662 p^{2} T^{7} + 400 p^{3} T^{8} - 21 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
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 \( ( 1 - 15 T + 359 T^{2} - 3943 T^{3} + 53173 T^{4} - 414929 T^{5} + 53173 p T^{6} - 3943 p^{2} T^{7} + 359 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
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 \( ( 1 - 28 T + 680 T^{2} - 10388 T^{3} + 140263 T^{4} - 1402827 T^{5} + 140263 p T^{6} - 10388 p^{2} T^{7} + 680 p^{3} T^{8} - 28 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
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

−4.10096284721631459774211876902, −4.06272244480376226873089402571, −3.84172644487950478505789308201, −3.59362019363070514481821934305, −3.57791503827437694115003077393, −3.54152298639352697267958007160, −3.38059629205463086175306585614, −3.37831316458771277531297714115, −3.34579123972321004116346480109, −3.21157133010579593866687050714, −3.06836368120998322368924491319, −3.05712346734765522272753855155, −2.88314655155537217798899518053, −2.74861061824448536008079892529, −2.36928308147583675035593101476, −2.21528388608458577474396997482, −2.13595004979518976371711111142, −1.96792868100166654967151973111, −1.86509311907150837187415458838, −1.26335148749866537888469935672, −1.22260651346530075232543579688, −1.17278024141915365176269407644, −1.07693927570911668138385012274, −0.910934178483642533455547964147, −0.69268386286219370261094706363, 0.69268386286219370261094706363, 0.910934178483642533455547964147, 1.07693927570911668138385012274, 1.17278024141915365176269407644, 1.22260651346530075232543579688, 1.26335148749866537888469935672, 1.86509311907150837187415458838, 1.96792868100166654967151973111, 2.13595004979518976371711111142, 2.21528388608458577474396997482, 2.36928308147583675035593101476, 2.74861061824448536008079892529, 2.88314655155537217798899518053, 3.05712346734765522272753855155, 3.06836368120998322368924491319, 3.21157133010579593866687050714, 3.34579123972321004116346480109, 3.37831316458771277531297714115, 3.38059629205463086175306585614, 3.54152298639352697267958007160, 3.57791503827437694115003077393, 3.59362019363070514481821934305, 3.84172644487950478505789308201, 4.06272244480376226873089402571, 4.10096284721631459774211876902

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

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