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Properties

Label 20-2034e10-1.1-c1e10-0-1
Degree $20$
Conductor $1.212\times 10^{33}$
Sign $1$
Analytic cond. $1.27726\times 10^{12}$
Root an. cond. $4.03008$
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 + 220·8-s + 2·11-s + 2·13-s + 715·16-s + 20·22-s + 25·25-s + 20·26-s − 8·31-s + 2.00e3·32-s − 8·41-s + 110·44-s − 34·49-s + 250·50-s + 110·52-s + 18·53-s + 2·61-s − 80·62-s + 5.00e3·64-s − 80·82-s − 14·83-s + 440·88-s + 10·97-s − 340·98-s + 1.37e3·100-s + 440·104-s + ⋯
L(s)  = 1  + 7.07·2-s + 55/2·4-s + 77.7·8-s + 0.603·11-s + 0.554·13-s + 178.·16-s + 4.26·22-s + 5·25-s + 3.92·26-s − 1.43·31-s + 353.·32-s − 1.24·41-s + 16.5·44-s − 4.85·49-s + 35.3·50-s + 15.2·52-s + 2.47·53-s + 0.256·61-s − 10.1·62-s + 625.·64-s − 8.83·82-s − 1.53·83-s + 46.9·88-s + 1.01·97-s − 34.3·98-s + 137.5·100-s + 43.1·104-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{20} \cdot 113^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{20} \cdot 113^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{10} \cdot 3^{20} \cdot 113^{10}\)
Sign: $1$
Analytic conductor: \(1.27726\times 10^{12}\)
Root analytic conductor: \(4.03008\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{10} \cdot 3^{20} \cdot 113^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1827.674792\)
\(L(\frac12)\) \(\approx\) \(1827.674792\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T )^{10} \)
3 \( 1 \)
113 \( 1 + 24 T + 217 T^{2} + 48 T^{3} - 36938 T^{4} - 622800 T^{5} - 36938 p T^{6} + 48 p^{2} T^{7} + 217 p^{3} T^{8} + 24 p^{4} T^{9} + p^{5} T^{10} \)
good5 \( 1 - p^{2} T^{2} + 63 p T^{4} - 2718 T^{6} + 18228 T^{8} - 100098 T^{10} + 18228 p^{2} T^{12} - 2718 p^{4} T^{14} + 63 p^{7} T^{16} - p^{10} T^{18} + p^{10} T^{20} \)
7 \( ( 1 + 17 T^{2} - 5 T^{3} + 184 T^{4} - 22 T^{5} + 184 p T^{6} - 5 p^{2} T^{7} + 17 p^{3} T^{8} + p^{5} T^{10} )^{2} \)
11 \( ( 1 - T + 20 T^{2} - 4 p T^{3} + 325 T^{4} - 618 T^{5} + 325 p T^{6} - 4 p^{3} T^{7} + 20 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \)
13 \( ( 1 - T + 27 T^{2} - 36 T^{3} + 408 T^{4} - 486 T^{5} + 408 p T^{6} - 36 p^{2} T^{7} + 27 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \)
17 \( 1 - 94 T^{2} + 3921 T^{4} - 98667 T^{6} + 1814826 T^{8} - 30340038 T^{10} + 1814826 p^{2} T^{12} - 98667 p^{4} T^{14} + 3921 p^{6} T^{16} - 94 p^{8} T^{18} + p^{10} T^{20} \)
19 \( 1 - 99 T^{2} + 5006 T^{4} - 166886 T^{6} + 4204321 T^{8} - 86720206 T^{10} + 4204321 p^{2} T^{12} - 166886 p^{4} T^{14} + 5006 p^{6} T^{16} - 99 p^{8} T^{18} + p^{10} T^{20} \)
23 \( 1 - 122 T^{2} + 6759 T^{4} - 227623 T^{6} + 5576600 T^{8} - 124712892 T^{10} + 5576600 p^{2} T^{12} - 227623 p^{4} T^{14} + 6759 p^{6} T^{16} - 122 p^{8} T^{18} + p^{10} T^{20} \)
29 \( 1 - 178 T^{2} + 15615 T^{4} - 903036 T^{6} + 38428536 T^{8} - 1260796068 T^{10} + 38428536 p^{2} T^{12} - 903036 p^{4} T^{14} + 15615 p^{6} T^{16} - 178 p^{8} T^{18} + p^{10} T^{20} \)
31 \( ( 1 + 4 T + 111 T^{2} + 407 T^{3} + 5918 T^{4} + 17994 T^{5} + 5918 p T^{6} + 407 p^{2} T^{7} + 111 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
37 \( 1 - 99 T^{2} + 6752 T^{4} - 389744 T^{6} + 18292345 T^{8} - 730992406 T^{10} + 18292345 p^{2} T^{12} - 389744 p^{4} T^{14} + 6752 p^{6} T^{16} - 99 p^{8} T^{18} + p^{10} T^{20} \)
41 \( ( 1 + 4 T + 95 T^{2} + 152 T^{3} + 4792 T^{4} + 3336 T^{5} + 4792 p T^{6} + 152 p^{2} T^{7} + 95 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
43 \( 1 - 51 T^{2} + 1322 T^{4} - 29930 T^{6} + 4647733 T^{8} - 354866758 T^{10} + 4647733 p^{2} T^{12} - 29930 p^{4} T^{14} + 1322 p^{6} T^{16} - 51 p^{8} T^{18} + p^{10} T^{20} \)
47 \( 1 - 331 T^{2} + 53292 T^{4} - 5481480 T^{6} + 399799893 T^{8} - 21655587318 T^{10} + 399799893 p^{2} T^{12} - 5481480 p^{4} T^{14} + 53292 p^{6} T^{16} - 331 p^{8} T^{18} + p^{10} T^{20} \)
53 \( ( 1 - 9 T + 88 T^{2} - 288 T^{3} + 6751 T^{4} - 43470 T^{5} + 6751 p T^{6} - 288 p^{2} T^{7} + 88 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
59 \( 1 - 289 T^{2} + 39909 T^{4} - 3371772 T^{6} + 206520498 T^{8} - 11632994214 T^{10} + 206520498 p^{2} T^{12} - 3371772 p^{4} T^{14} + 39909 p^{6} T^{16} - 289 p^{8} T^{18} + p^{10} T^{20} \)
61 \( ( 1 - T + 267 T^{2} - 228 T^{3} + 30456 T^{4} - 20262 T^{5} + 30456 p T^{6} - 228 p^{2} T^{7} + 267 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \)
67 \( 1 - 183 T^{2} + 31430 T^{4} - 3408950 T^{6} + 322290025 T^{8} - 23227538566 T^{10} + 322290025 p^{2} T^{12} - 3408950 p^{4} T^{14} + 31430 p^{6} T^{16} - 183 p^{8} T^{18} + p^{10} T^{20} \)
71 \( 1 - 454 T^{2} + 99123 T^{4} - 13942503 T^{6} + 1430560164 T^{8} - 114051239220 T^{10} + 1430560164 p^{2} T^{12} - 13942503 p^{4} T^{14} + 99123 p^{6} T^{16} - 454 p^{8} T^{18} + p^{10} T^{20} \)
73 \( 1 - 126 T^{2} + 19853 T^{4} - 2042504 T^{6} + 187343170 T^{8} - 15385552756 T^{10} + 187343170 p^{2} T^{12} - 2042504 p^{4} T^{14} + 19853 p^{6} T^{16} - 126 p^{8} T^{18} + p^{10} T^{20} \)
79 \( 1 - 330 T^{2} + 50183 T^{4} - 4185212 T^{6} + 199263544 T^{8} - 8677975732 T^{10} + 199263544 p^{2} T^{12} - 4185212 p^{4} T^{14} + 50183 p^{6} T^{16} - 330 p^{8} T^{18} + p^{10} T^{20} \)
83 \( ( 1 + 7 T + 212 T^{2} + 2306 T^{3} + 24877 T^{4} + 275766 T^{5} + 24877 p T^{6} + 2306 p^{2} T^{7} + 212 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
89 \( 1 - 487 T^{2} + 123906 T^{4} - 21346446 T^{6} + 2744502141 T^{8} - 275103020310 T^{10} + 2744502141 p^{2} T^{12} - 21346446 p^{4} T^{14} + 123906 p^{6} T^{16} - 487 p^{8} T^{18} + p^{10} T^{20} \)
97 \( ( 1 - 5 T + 264 T^{2} - 1888 T^{3} + 32465 T^{4} - 274650 T^{5} + 32465 p T^{6} - 1888 p^{2} T^{7} + 264 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
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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

−3.39450093463639182122679742165, −3.36809694613050610844591935292, −3.09172067202817347788906542366, −2.99916848576938169127297989415, −2.84009876706613794354577228870, −2.83059795887037255231404993511, −2.80452279083582692682827036533, −2.80401802112634282827346543678, −2.55713441492918321199821186960, −2.44895452648305761650277832705, −2.41422636937817928100301349482, −2.15885868118422871402319694801, −2.14434967533169998940293044451, −2.03039281902100178525800764842, −1.82347594533628231856863260808, −1.81525759644642049799342248057, −1.57347306022378700909351759608, −1.37640595930819042394200435957, −1.37163600582904006680873769907, −1.30316443384326401879076099463, −1.17176305778321780327489438332, −1.07454219606424740680531417801, −0.813661129120093773244458251446, −0.53851834842876467024859554843, −0.18139956880761897272689606303, 0.18139956880761897272689606303, 0.53851834842876467024859554843, 0.813661129120093773244458251446, 1.07454219606424740680531417801, 1.17176305778321780327489438332, 1.30316443384326401879076099463, 1.37163600582904006680873769907, 1.37640595930819042394200435957, 1.57347306022378700909351759608, 1.81525759644642049799342248057, 1.82347594533628231856863260808, 2.03039281902100178525800764842, 2.14434967533169998940293044451, 2.15885868118422871402319694801, 2.41422636937817928100301349482, 2.44895452648305761650277832705, 2.55713441492918321199821186960, 2.80401802112634282827346543678, 2.80452279083582692682827036533, 2.83059795887037255231404993511, 2.84009876706613794354577228870, 2.99916848576938169127297989415, 3.09172067202817347788906542366, 3.36809694613050610844591935292, 3.39450093463639182122679742165

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

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