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Properties

Label 20-3330e10-1.1-c1e10-0-0
Degree $20$
Conductor $1.677\times 10^{35}$
Sign $1$
Analytic cond. $1.76690\times 10^{14}$
Root an. cond. $5.15656$
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 + 3·5-s + 220·8-s + 30·10-s − 2·13-s + 715·16-s − 18·17-s + 165·20-s − 10·23-s + 7·25-s − 20·26-s + 2.00e3·32-s − 180·34-s − 8·37-s + 660·40-s + 4·41-s − 10·43-s − 100·46-s + 31·49-s + 70·50-s − 110·52-s + 5.00e3·64-s − 6·65-s − 990·68-s + 20·71-s − 80·74-s + ⋯
L(s)  = 1  + 7.07·2-s + 55/2·4-s + 1.34·5-s + 77.7·8-s + 9.48·10-s − 0.554·13-s + 178.·16-s − 4.36·17-s + 36.8·20-s − 2.08·23-s + 7/5·25-s − 3.92·26-s + 353.·32-s − 30.8·34-s − 1.31·37-s + 104.·40-s + 0.624·41-s − 1.52·43-s − 14.7·46-s + 31/7·49-s + 9.89·50-s − 15.2·52-s + 625.·64-s − 0.744·65-s − 120.·68-s + 2.37·71-s − 9.29·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{20} \cdot 5^{10} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{20} \cdot 5^{10} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{10} \cdot 3^{20} \cdot 5^{10} \cdot 37^{10}\)
Sign: $1$
Analytic conductor: \(1.76690\times 10^{14}\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{10} \cdot 3^{20} \cdot 5^{10} \cdot 37^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1357.540080\)
\(L(\frac12)\) \(\approx\) \(1357.540080\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - 3 T + 2 T^{2} + 16 T^{3} - 19 T^{4} + 22 T^{5} - 19 p T^{6} + 16 p^{2} T^{7} + 2 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
37 \( 1 + 8 T + 97 T^{2} + 576 T^{3} + 5282 T^{4} + 25584 T^{5} + 5282 p T^{6} + 576 p^{2} T^{7} + 97 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
good7 \( 1 - 31 T^{2} + 459 T^{4} - 4364 T^{6} + 31924 T^{8} - 218314 T^{10} + 31924 p^{2} T^{12} - 4364 p^{4} T^{14} + 459 p^{6} T^{16} - 31 p^{8} T^{18} + p^{10} T^{20} \)
11 \( ( 1 + 27 T^{2} - 51 T^{3} + 302 T^{4} - 1074 T^{5} + 302 p T^{6} - 51 p^{2} T^{7} + 27 p^{3} T^{8} + p^{5} T^{10} )^{2} \)
13 \( ( 1 + T + 2 p T^{2} - 48 T^{3} + 329 T^{4} - 1098 T^{5} + 329 p T^{6} - 48 p^{2} T^{7} + 2 p^{4} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \)
17 \( ( 1 + 9 T + 89 T^{2} + 504 T^{3} + 2982 T^{4} + 12078 T^{5} + 2982 p T^{6} + 504 p^{2} T^{7} + 89 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
19 \( 1 - 72 T^{2} + 2977 T^{4} - 90248 T^{6} + 2204454 T^{8} - 45625440 T^{10} + 2204454 p^{2} T^{12} - 90248 p^{4} T^{14} + 2977 p^{6} T^{16} - 72 p^{8} T^{18} + p^{10} T^{20} \)
23 \( ( 1 + 5 T + 52 T^{2} + 152 T^{3} + 1199 T^{4} + 2470 T^{5} + 1199 p T^{6} + 152 p^{2} T^{7} + 52 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
29 \( 1 - 108 T^{2} + 5879 T^{4} - 231107 T^{6} + 276700 p T^{8} - 249440874 T^{10} + 276700 p^{3} T^{12} - 231107 p^{4} T^{14} + 5879 p^{6} T^{16} - 108 p^{8} T^{18} + p^{10} T^{20} \)
31 \( 1 - 194 T^{2} + 18979 T^{4} - 1222911 T^{6} + 57220962 T^{8} - 2026943118 T^{10} + 57220962 p^{2} T^{12} - 1222911 p^{4} T^{14} + 18979 p^{6} T^{16} - 194 p^{8} T^{18} + p^{10} T^{20} \)
41 \( ( 1 - 2 T + 161 T^{2} - 417 T^{3} + 11390 T^{4} - 27434 T^{5} + 11390 p T^{6} - 417 p^{2} T^{7} + 161 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
43 \( ( 1 + 5 T + 127 T^{2} + 784 T^{3} + 8994 T^{4} + 46310 T^{5} + 8994 p T^{6} + 784 p^{2} T^{7} + 127 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
47 \( 1 - 160 T^{2} + 11993 T^{4} - 511560 T^{6} + 12029318 T^{8} - 253577648 T^{10} + 12029318 p^{2} T^{12} - 511560 p^{4} T^{14} + 11993 p^{6} T^{16} - 160 p^{8} T^{18} + p^{10} T^{20} \)
53 \( 1 - 273 T^{2} + 40401 T^{4} - 4128228 T^{6} + 317307718 T^{8} - 18960642934 T^{10} + 317307718 p^{2} T^{12} - 4128228 p^{4} T^{14} + 40401 p^{6} T^{16} - 273 p^{8} T^{18} + p^{10} T^{20} \)
59 \( 1 - 352 T^{2} + 63153 T^{4} - 7528424 T^{6} + 659017222 T^{8} - 44169143152 T^{10} + 659017222 p^{2} T^{12} - 7528424 p^{4} T^{14} + 63153 p^{6} T^{16} - 352 p^{8} T^{18} + p^{10} T^{20} \)
61 \( 1 - 460 T^{2} + 97647 T^{4} - 12826483 T^{6} + 19411236 p T^{8} - 82282695178 T^{10} + 19411236 p^{3} T^{12} - 12826483 p^{4} T^{14} + 97647 p^{6} T^{16} - 460 p^{8} T^{18} + p^{10} T^{20} \)
67 \( 1 - 303 T^{2} + 54020 T^{4} - 6711456 T^{6} + 641201935 T^{8} - 48089295626 T^{10} + 641201935 p^{2} T^{12} - 6711456 p^{4} T^{14} + 54020 p^{6} T^{16} - 303 p^{8} T^{18} + p^{10} T^{20} \)
71 \( ( 1 - 10 T + 191 T^{2} - 944 T^{3} + 12662 T^{4} - 37836 T^{5} + 12662 p T^{6} - 944 p^{2} T^{7} + 191 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
73 \( 1 - 541 T^{2} + 140394 T^{4} - 23094490 T^{6} + 2668314981 T^{8} - 226082874802 T^{10} + 2668314981 p^{2} T^{12} - 23094490 p^{4} T^{14} + 140394 p^{6} T^{16} - 541 p^{8} T^{18} + p^{10} T^{20} \)
79 \( 1 - 499 T^{2} + 127952 T^{4} - 21543420 T^{6} + 2613057515 T^{8} - 237224216810 T^{10} + 2613057515 p^{2} T^{12} - 21543420 p^{4} T^{14} + 127952 p^{6} T^{16} - 499 p^{8} T^{18} + p^{10} T^{20} \)
83 \( 1 - 364 T^{2} + 62793 T^{4} - 7248824 T^{6} + 680746366 T^{8} - 58266464440 T^{10} + 680746366 p^{2} T^{12} - 7248824 p^{4} T^{14} + 62793 p^{6} T^{16} - 364 p^{8} T^{18} + p^{10} T^{20} \)
89 \( 1 - 478 T^{2} + 122989 T^{4} - 21250184 T^{6} + 2738250882 T^{8} - 274126598068 T^{10} + 2738250882 p^{2} T^{12} - 21250184 p^{4} T^{14} + 122989 p^{6} T^{16} - 478 p^{8} T^{18} + p^{10} T^{20} \)
97 \( ( 1 - T + 307 T^{2} - 636 T^{3} + 45116 T^{4} - 103398 T^{5} + 45116 p T^{6} - 636 p^{2} T^{7} + 307 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \)
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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

−3.13728032284201073941402917340, −2.94828568853183931613528253850, −2.87744997288477787581140847562, −2.72010758597315014577286701361, −2.66727540574760412885472075663, −2.59261509869265145186528654973, −2.41707278915983049529952228012, −2.31712662621110413725884311790, −2.23035949298875941565291446566, −2.22186176257525086215590071402, −2.21223812875443969815376098001, −2.17839496552638751684542141293, −2.07006144258782099090492130265, −1.82936935063582624333291398405, −1.81446846346585137559333345416, −1.75161809399767387298981716949, −1.69110611784907131820014590725, −1.26323913287837499020987330199, −1.25690459554832312652526688955, −1.24365508118008097060699086708, −0.992420230554826953044872328337, −0.797996050584092551686557668841, −0.57494781065377689829316661191, −0.44187610781905643703390010776, −0.13681034262919523107425072641, 0.13681034262919523107425072641, 0.44187610781905643703390010776, 0.57494781065377689829316661191, 0.797996050584092551686557668841, 0.992420230554826953044872328337, 1.24365508118008097060699086708, 1.25690459554832312652526688955, 1.26323913287837499020987330199, 1.69110611784907131820014590725, 1.75161809399767387298981716949, 1.81446846346585137559333345416, 1.82936935063582624333291398405, 2.07006144258782099090492130265, 2.17839496552638751684542141293, 2.21223812875443969815376098001, 2.22186176257525086215590071402, 2.23035949298875941565291446566, 2.31712662621110413725884311790, 2.41707278915983049529952228012, 2.59261509869265145186528654973, 2.66727540574760412885472075663, 2.72010758597315014577286701361, 2.87744997288477787581140847562, 2.94828568853183931613528253850, 3.13728032284201073941402917340

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

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