Properties

Label 20-2520e10-1.1-c1e10-0-1
Degree $20$
Conductor $1.033\times 10^{34}$
Sign $1$
Analytic cond. $1.08836\times 10^{13}$
Root an. cond. $4.48578$
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  − 5·5-s − 7-s − 2·11-s + 6·13-s + 2·17-s + 19-s + 8·23-s + 10·25-s + 7·31-s + 5·35-s − 11·37-s + 20·41-s + 6·43-s − 11·49-s − 14·53-s + 10·55-s + 4·59-s − 6·61-s − 30·65-s − 7·67-s − 32·71-s + 3·73-s + 2·77-s − 19·79-s + 28·83-s − 10·85-s − 18·89-s + ⋯
L(s)  = 1  − 2.23·5-s − 0.377·7-s − 0.603·11-s + 1.66·13-s + 0.485·17-s + 0.229·19-s + 1.66·23-s + 2·25-s + 1.25·31-s + 0.845·35-s − 1.80·37-s + 3.12·41-s + 0.914·43-s − 1.57·49-s − 1.92·53-s + 1.34·55-s + 0.520·59-s − 0.768·61-s − 3.72·65-s − 0.855·67-s − 3.79·71-s + 0.351·73-s + 0.227·77-s − 2.13·79-s + 3.07·83-s − 1.08·85-s − 1.90·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{20} \cdot 5^{10} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{20} \cdot 5^{10} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{30} \cdot 3^{20} \cdot 5^{10} \cdot 7^{10}\)
Sign: $1$
Analytic conductor: \(1.08836\times 10^{13}\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2520} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{30} \cdot 3^{20} \cdot 5^{10} \cdot 7^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(16.71769201\)
\(L(\frac12)\) \(\approx\) \(16.71769201\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( ( 1 + T + T^{2} )^{5} \)
7 \( 1 + T + 12 T^{2} + 5 p T^{3} + 59 T^{4} + 372 T^{5} + 59 p T^{6} + 5 p^{3} T^{7} + 12 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
good11 \( 1 + 2 T - 10 T^{2} + 84 T^{3} + 177 T^{4} - 102 p T^{5} + 204 p T^{6} + 8202 T^{7} - 50151 T^{8} - 54778 T^{9} + 330298 T^{10} - 54778 p T^{11} - 50151 p^{2} T^{12} + 8202 p^{3} T^{13} + 204 p^{5} T^{14} - 102 p^{6} T^{15} + 177 p^{6} T^{16} + 84 p^{7} T^{17} - 10 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \)
13 \( ( 1 - 3 T + 23 T^{2} - 112 T^{3} + 577 T^{4} - 1427 T^{5} + 577 p T^{6} - 112 p^{2} T^{7} + 23 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
17 \( 1 - 2 T - 13 T^{2} + 42 T^{3} - 273 T^{4} + 474 T^{5} + 474 p T^{6} - 17958 T^{7} + 1461 p T^{8} + 207796 T^{9} - 2667899 T^{10} + 207796 p T^{11} + 1461 p^{3} T^{12} - 17958 p^{3} T^{13} + 474 p^{5} T^{14} + 474 p^{5} T^{15} - 273 p^{6} T^{16} + 42 p^{7} T^{17} - 13 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \)
19 \( 1 - T - 32 T^{2} - 237 T^{3} + 1014 T^{4} + 8325 T^{5} + 15894 T^{6} - 252417 T^{7} - 849351 T^{8} + 1378454 T^{9} + 29187412 T^{10} + 1378454 p T^{11} - 849351 p^{2} T^{12} - 252417 p^{3} T^{13} + 15894 p^{4} T^{14} + 8325 p^{5} T^{15} + 1014 p^{6} T^{16} - 237 p^{7} T^{17} - 32 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \)
23 \( 1 - 8 T - 4 T^{2} + 104 T^{3} + 593 T^{4} - 5546 T^{5} + 21014 T^{6} - 13240 T^{7} - 383207 T^{8} - 1757646 T^{9} + 27760806 T^{10} - 1757646 p T^{11} - 383207 p^{2} T^{12} - 13240 p^{3} T^{13} + 21014 p^{4} T^{14} - 5546 p^{5} T^{15} + 593 p^{6} T^{16} + 104 p^{7} T^{17} - 4 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \)
29 \( ( 1 + 55 T^{2} - 46 T^{3} + 1540 T^{4} - 3052 T^{5} + 1540 p T^{6} - 46 p^{2} T^{7} + 55 p^{3} T^{8} + p^{5} T^{10} )^{2} \)
31 \( 1 - 7 T - 5 T^{2} + 768 T^{3} - 4728 T^{4} + 2388 T^{5} + 234669 T^{6} - 1527267 T^{7} + 3171639 T^{8} + 37305236 T^{9} - 317453072 T^{10} + 37305236 p T^{11} + 3171639 p^{2} T^{12} - 1527267 p^{3} T^{13} + 234669 p^{4} T^{14} + 2388 p^{5} T^{15} - 4728 p^{6} T^{16} + 768 p^{7} T^{17} - 5 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \)
37 \( 1 + 11 T - 80 T^{2} - 865 T^{3} + 8074 T^{4} + 53837 T^{5} - 512166 T^{6} - 1570167 T^{7} + 30254793 T^{8} + 29930128 T^{9} - 1221538156 T^{10} + 29930128 p T^{11} + 30254793 p^{2} T^{12} - 1570167 p^{3} T^{13} - 512166 p^{4} T^{14} + 53837 p^{5} T^{15} + 8074 p^{6} T^{16} - 865 p^{7} T^{17} - 80 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \)
41 \( ( 1 - 10 T + 152 T^{2} - 920 T^{3} + 8533 T^{4} - 41322 T^{5} + 8533 p T^{6} - 920 p^{2} T^{7} + 152 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
43 \( ( 1 - 3 T + 74 T^{2} + 5 T^{3} + 4057 T^{4} - 2900 T^{5} + 4057 p T^{6} + 5 p^{2} T^{7} + 74 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
47 \( 1 - 124 T^{2} + 268 T^{3} + 8685 T^{4} - 28996 T^{5} - 182774 T^{6} + 1877640 T^{7} - 8758639 T^{8} - 35259488 T^{9} + 972853878 T^{10} - 35259488 p T^{11} - 8758639 p^{2} T^{12} + 1877640 p^{3} T^{13} - 182774 p^{4} T^{14} - 28996 p^{5} T^{15} + 8685 p^{6} T^{16} + 268 p^{7} T^{17} - 124 p^{8} T^{18} + p^{10} T^{20} \)
53 \( 1 + 14 T - 82 T^{2} - 1324 T^{3} + 11431 T^{4} + 100936 T^{5} - 1009428 T^{6} - 4169538 T^{7} + 77344917 T^{8} + 77501324 T^{9} - 4695929102 T^{10} + 77501324 p T^{11} + 77344917 p^{2} T^{12} - 4169538 p^{3} T^{13} - 1009428 p^{4} T^{14} + 100936 p^{5} T^{15} + 11431 p^{6} T^{16} - 1324 p^{7} T^{17} - 82 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \)
59 \( 1 - 4 T - 193 T^{2} + 8 p T^{3} + 20573 T^{4} - 25042 T^{5} - 1646224 T^{6} + 1343602 T^{7} + 108324625 T^{8} - 40720962 T^{9} - 6496309785 T^{10} - 40720962 p T^{11} + 108324625 p^{2} T^{12} + 1343602 p^{3} T^{13} - 1646224 p^{4} T^{14} - 25042 p^{5} T^{15} + 20573 p^{6} T^{16} + 8 p^{8} T^{17} - 193 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \)
61 \( 1 + 6 T - 101 T^{2} - 290 T^{3} + 6999 T^{4} - 1184 T^{5} + 39514 T^{6} + 3309576 T^{7} - 18882179 T^{8} - 67063018 T^{9} + 2256915333 T^{10} - 67063018 p T^{11} - 18882179 p^{2} T^{12} + 3309576 p^{3} T^{13} + 39514 p^{4} T^{14} - 1184 p^{5} T^{15} + 6999 p^{6} T^{16} - 290 p^{7} T^{17} - 101 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \)
67 \( 1 + 7 T - 95 T^{2} + 102 T^{3} + 10110 T^{4} - 30426 T^{5} + 69675 T^{6} + 6397971 T^{7} - 35135985 T^{8} - 73705592 T^{9} + 5566768168 T^{10} - 73705592 p T^{11} - 35135985 p^{2} T^{12} + 6397971 p^{3} T^{13} + 69675 p^{4} T^{14} - 30426 p^{5} T^{15} + 10110 p^{6} T^{16} + 102 p^{7} T^{17} - 95 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \)
71 \( ( 1 + 16 T + 365 T^{2} + 4058 T^{3} + 51892 T^{4} + 414708 T^{5} + 51892 p T^{6} + 4058 p^{2} T^{7} + 365 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
73 \( 1 - 3 T - 131 T^{2} - 1326 T^{3} + 11472 T^{4} + 185946 T^{5} + 458943 T^{6} - 13735125 T^{7} - 105619185 T^{8} + 339901428 T^{9} + 9560905356 T^{10} + 339901428 p T^{11} - 105619185 p^{2} T^{12} - 13735125 p^{3} T^{13} + 458943 p^{4} T^{14} + 185946 p^{5} T^{15} + 11472 p^{6} T^{16} - 1326 p^{7} T^{17} - 131 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \)
79 \( 1 + 19 T + 91 T^{2} + 592 T^{3} + 17944 T^{4} + 248416 T^{5} + 2189697 T^{6} + 12432639 T^{7} + 97743123 T^{8} + 1643592752 T^{9} + 18910234400 T^{10} + 1643592752 p T^{11} + 97743123 p^{2} T^{12} + 12432639 p^{3} T^{13} + 2189697 p^{4} T^{14} + 248416 p^{5} T^{15} + 17944 p^{6} T^{16} + 592 p^{7} T^{17} + 91 p^{8} T^{18} + 19 p^{9} T^{19} + p^{10} T^{20} \)
83 \( ( 1 - 14 T + 179 T^{2} - 1266 T^{3} + 22502 T^{4} - 203392 T^{5} + 22502 p T^{6} - 1266 p^{2} T^{7} + 179 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
89 \( 1 + 18 T - 163 T^{2} - 3678 T^{3} + 35469 T^{4} + 622554 T^{5} - 3797364 T^{6} - 49289058 T^{7} + 476547825 T^{8} + 2291267244 T^{9} - 39003817743 T^{10} + 2291267244 p T^{11} + 476547825 p^{2} T^{12} - 49289058 p^{3} T^{13} - 3797364 p^{4} T^{14} + 622554 p^{5} T^{15} + 35469 p^{6} T^{16} - 3678 p^{7} T^{17} - 163 p^{8} T^{18} + 18 p^{9} T^{19} + p^{10} T^{20} \)
97 \( ( 1 - 24 T + 5 p T^{2} - 6388 T^{3} + 84202 T^{4} - 838136 T^{5} + 84202 p T^{6} - 6388 p^{2} T^{7} + 5 p^{4} T^{8} - 24 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
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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

−3.03950519460009906898456087830, −3.00587893397950284253884236339, −3.00035689121528198588591202858, −2.99548551917208360749376292488, −2.91844887871293588357542268236, −2.69028295517130489506848438548, −2.60219077953659596950827113074, −2.44413649616669743096871901892, −2.39668902500673108158257958064, −2.21420759797314944021278863539, −2.10442954384497139541344240726, −1.89239289817448895109496838354, −1.80234991774371805099950164739, −1.65490205935222779477684947821, −1.64678374875792205053776119784, −1.44732781803402047033211546011, −1.35703925025876931329513286325, −1.29442023127373395396890083953, −1.19014659850874504236755338937, −0.825852730327768804774244217780, −0.62127155595022370120142160993, −0.54847839815438041346020002685, −0.47049242538506832725394069240, −0.42873652321206275857672553592, −0.41515399602024727943256816733, 0.41515399602024727943256816733, 0.42873652321206275857672553592, 0.47049242538506832725394069240, 0.54847839815438041346020002685, 0.62127155595022370120142160993, 0.825852730327768804774244217780, 1.19014659850874504236755338937, 1.29442023127373395396890083953, 1.35703925025876931329513286325, 1.44732781803402047033211546011, 1.64678374875792205053776119784, 1.65490205935222779477684947821, 1.80234991774371805099950164739, 1.89239289817448895109496838354, 2.10442954384497139541344240726, 2.21420759797314944021278863539, 2.39668902500673108158257958064, 2.44413649616669743096871901892, 2.60219077953659596950827113074, 2.69028295517130489506848438548, 2.91844887871293588357542268236, 2.99548551917208360749376292488, 3.00035689121528198588591202858, 3.00587893397950284253884236339, 3.03950519460009906898456087830

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

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