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Properties

Label 20-4368e10-1.1-c1e10-0-1
Degree $20$
Conductor $2.528\times 10^{36}$
Sign $1$
Analytic cond. $2.66438\times 10^{15}$
Root an. cond. $5.90581$
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  − 10·3-s + 55·9-s − 4·13-s + 10·17-s − 8·23-s + 24·25-s − 220·27-s − 44·29-s + 40·39-s − 20·43-s − 5·49-s − 100·51-s + 10·53-s + 18·61-s + 80·69-s − 240·75-s − 2·79-s + 715·81-s + 440·87-s + 60·101-s − 6·103-s + 8·107-s − 10·113-s − 220·117-s + 37·121-s + 127-s + 200·129-s + ⋯
L(s)  = 1  − 5.77·3-s + 55/3·9-s − 1.10·13-s + 2.42·17-s − 1.66·23-s + 24/5·25-s − 42.3·27-s − 8.17·29-s + 6.40·39-s − 3.04·43-s − 5/7·49-s − 14.0·51-s + 1.37·53-s + 2.30·61-s + 9.63·69-s − 27.7·75-s − 0.225·79-s + 79.4·81-s + 47.1·87-s + 5.97·101-s − 0.591·103-s + 0.773·107-s − 0.940·113-s − 20.3·117-s + 3.36·121-s + 0.0887·127-s + 17.6·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{10} \cdot 7^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{10} \cdot 7^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{40} \cdot 3^{10} \cdot 7^{10} \cdot 13^{10}\)
Sign: $1$
Analytic conductor: \(2.66438\times 10^{15}\)
Root analytic conductor: \(5.90581\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{40} \cdot 3^{10} \cdot 7^{10} \cdot 13^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.05557309316\)
\(L(\frac12)\) \(\approx\) \(0.05557309316\)
\(L(\frac{3}{2})\) 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 \( ( 1 + T )^{10} \)
7 \( ( 1 + T^{2} )^{5} \)
13 \( 1 + 4 T + 33 T^{2} + 96 T^{3} + 586 T^{4} + 1336 T^{5} + 586 p T^{6} + 96 p^{2} T^{7} + 33 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
good5 \( 1 - 24 T^{2} + 334 T^{4} - 646 p T^{6} + 937 p^{2} T^{8} - 132484 T^{10} + 937 p^{4} T^{12} - 646 p^{5} T^{14} + 334 p^{6} T^{16} - 24 p^{8} T^{18} + p^{10} T^{20} \)
11 \( 1 - 37 T^{2} + 997 T^{4} - 18620 T^{6} + 284002 T^{8} - 3419390 T^{10} + 284002 p^{2} T^{12} - 18620 p^{4} T^{14} + 997 p^{6} T^{16} - 37 p^{8} T^{18} + p^{10} T^{20} \)
17 \( ( 1 - 5 T + 59 T^{2} - 236 T^{3} + 1660 T^{4} - 5518 T^{5} + 1660 p T^{6} - 236 p^{2} T^{7} + 59 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
19 \( 1 - 116 T^{2} + 6398 T^{4} - 226562 T^{6} + 5909521 T^{8} - 123468500 T^{10} + 5909521 p^{2} T^{12} - 226562 p^{4} T^{14} + 6398 p^{6} T^{16} - 116 p^{8} T^{18} + p^{10} T^{20} \)
23 \( ( 1 + 4 T + 28 T^{2} - 94 T^{3} + 91 T^{4} - 3284 T^{5} + 91 p T^{6} - 94 p^{2} T^{7} + 28 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
29 \( ( 1 + 22 T + 248 T^{2} + 2014 T^{3} + 13663 T^{4} + 79592 T^{5} + 13663 p T^{6} + 2014 p^{2} T^{7} + 248 p^{3} T^{8} + 22 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
31 \( 1 - 149 T^{2} + 11573 T^{4} - 598124 T^{6} + 23644618 T^{8} - 783769214 T^{10} + 23644618 p^{2} T^{12} - 598124 p^{4} T^{14} + 11573 p^{6} T^{16} - 149 p^{8} T^{18} + p^{10} T^{20} \)
37 \( 1 - 297 T^{2} + 41245 T^{4} - 3548988 T^{6} + 210941114 T^{8} - 9096299158 T^{10} + 210941114 p^{2} T^{12} - 3548988 p^{4} T^{14} + 41245 p^{6} T^{16} - 297 p^{8} T^{18} + p^{10} T^{20} \)
41 \( 1 - 86 T^{2} + 9821 T^{4} - 558008 T^{6} + 35076658 T^{8} - 1392556388 T^{10} + 35076658 p^{2} T^{12} - 558008 p^{4} T^{14} + 9821 p^{6} T^{16} - 86 p^{8} T^{18} + p^{10} T^{20} \)
43 \( ( 1 + 10 T + 140 T^{2} + 1240 T^{3} + 11119 T^{4} + 67612 T^{5} + 11119 p T^{6} + 1240 p^{2} T^{7} + 140 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
47 \( 1 + 23 T^{2} + 3721 T^{4} - 108452 T^{6} - 171482 T^{8} - 701464646 T^{10} - 171482 p^{2} T^{12} - 108452 p^{4} T^{14} + 3721 p^{6} T^{16} + 23 p^{8} T^{18} + p^{10} T^{20} \)
53 \( ( 1 - 5 T + 177 T^{2} - 1132 T^{3} + 15138 T^{4} - 89870 T^{5} + 15138 p T^{6} - 1132 p^{2} T^{7} + 177 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
59 \( 1 - 230 T^{2} + 28325 T^{4} - 2471352 T^{6} + 176049010 T^{8} - 10926839300 T^{10} + 176049010 p^{2} T^{12} - 2471352 p^{4} T^{14} + 28325 p^{6} T^{16} - 230 p^{8} T^{18} + p^{10} T^{20} \)
61 \( ( 1 - 9 T + 247 T^{2} - 1824 T^{3} + 26780 T^{4} - 156862 T^{5} + 26780 p T^{6} - 1824 p^{2} T^{7} + 247 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
67 \( 1 + 6 T^{2} + 9781 T^{4} + 252360 T^{6} + 31376210 T^{8} + 2018659364 T^{10} + 31376210 p^{2} T^{12} + 252360 p^{4} T^{14} + 9781 p^{6} T^{16} + 6 p^{8} T^{18} + p^{10} T^{20} \)
71 \( 1 - 506 T^{2} + 123917 T^{4} - 19399112 T^{6} + 2150846242 T^{8} - 176432662268 T^{10} + 2150846242 p^{2} T^{12} - 19399112 p^{4} T^{14} + 123917 p^{6} T^{16} - 506 p^{8} T^{18} + p^{10} T^{20} \)
73 \( 1 - 544 T^{2} + 143990 T^{4} - 24150582 T^{6} + 2826549529 T^{8} - 240555273140 T^{10} + 2826549529 p^{2} T^{12} - 24150582 p^{4} T^{14} + 143990 p^{6} T^{16} - 544 p^{8} T^{18} + p^{10} T^{20} \)
79 \( ( 1 + T + 139 T^{2} - 364 T^{3} + 14938 T^{4} - 34858 T^{5} + 14938 p T^{6} - 364 p^{2} T^{7} + 139 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \)
83 \( 1 - 117 T^{2} + 2885 T^{4} - 158028 T^{6} + 49014194 T^{8} - 4228579646 T^{10} + 49014194 p^{2} T^{12} - 158028 p^{4} T^{14} + 2885 p^{6} T^{16} - 117 p^{8} T^{18} + p^{10} T^{20} \)
89 \( 1 - 417 T^{2} + 83909 T^{4} - 11608428 T^{6} + 1304843834 T^{8} - 125465819846 T^{10} + 1304843834 p^{2} T^{12} - 11608428 p^{4} T^{14} + 83909 p^{6} T^{16} - 417 p^{8} T^{18} + p^{10} T^{20} \)
97 \( 1 - 213 T^{2} + 47913 T^{4} - 7404292 T^{6} + 904237974 T^{8} - 101897520462 T^{10} + 904237974 p^{2} T^{12} - 7404292 p^{4} T^{14} + 47913 p^{6} T^{16} - 213 p^{8} T^{18} + p^{10} T^{20} \)
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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

−2.97923486112513358962720117705, −2.70622566941817204109084859185, −2.64179102220851428215339724228, −2.45699294592085218125014770413, −2.38904199142704817899262656298, −2.37669195659183875743466427484, −2.15348446846338414219685144633, −2.05888928439312657119425577133, −2.02492250266142926091235973838, −1.97473754530438484671899792868, −1.82261721649861788513397417736, −1.70774312927512662094669414645, −1.51409378809102503469499761096, −1.48827150948788378715861086557, −1.42590180619301654373035290517, −1.22984718333357153357698209553, −1.15824730248208900420687380671, −1.00126814838472669839988258618, −0.874730061263843361877109104911, −0.823692777353781997354253600857, −0.72544773759919103853115234676, −0.43409193054771208746528764470, −0.27664124364277934690892309444, −0.19685385697536426192569054037, −0.085343872021307543710368641923, 0.085343872021307543710368641923, 0.19685385697536426192569054037, 0.27664124364277934690892309444, 0.43409193054771208746528764470, 0.72544773759919103853115234676, 0.823692777353781997354253600857, 0.874730061263843361877109104911, 1.00126814838472669839988258618, 1.15824730248208900420687380671, 1.22984718333357153357698209553, 1.42590180619301654373035290517, 1.48827150948788378715861086557, 1.51409378809102503469499761096, 1.70774312927512662094669414645, 1.82261721649861788513397417736, 1.97473754530438484671899792868, 2.02492250266142926091235973838, 2.05888928439312657119425577133, 2.15348446846338414219685144633, 2.37669195659183875743466427484, 2.38904199142704817899262656298, 2.45699294592085218125014770413, 2.64179102220851428215339724228, 2.70622566941817204109084859185, 2.97923486112513358962720117705

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

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