Properties

Label 20-520e10-1.1-c1e10-0-0
Degree $20$
Conductor $1.446\times 10^{27}$
Sign $1$
Analytic cond. $1.52336\times 10^{6}$
Root an. cond. $2.03769$
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  − 4·5-s + 16·11-s − 36·19-s + 14·25-s − 4·29-s − 8·31-s − 24·41-s + 16·49-s − 64·55-s − 28·59-s + 20·61-s + 36·71-s − 16·79-s − 18·81-s + 12·89-s + 144·95-s + 44·101-s − 88·109-s + 122·121-s − 32·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1.78·5-s + 4.82·11-s − 8.25·19-s + 14/5·25-s − 0.742·29-s − 1.43·31-s − 3.74·41-s + 16/7·49-s − 8.62·55-s − 3.64·59-s + 2.56·61-s + 4.27·71-s − 1.80·79-s − 2·81-s + 1.27·89-s + 14.7·95-s + 4.37·101-s − 8.42·109-s + 11.0·121-s − 2.86·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 5^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 5^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{30} \cdot 5^{10} \cdot 13^{10}\)
Sign: $1$
Analytic conductor: \(1.52336\times 10^{6}\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{30} \cdot 5^{10} \cdot 13^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.092019544\)
\(L(\frac12)\) \(\approx\) \(1.092019544\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + 4 T + 2 T^{2} - 16 T^{3} - 3 p T^{4} + 8 p T^{5} - 3 p^{2} T^{6} - 16 p^{2} T^{7} + 2 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
13 \( ( 1 + T^{2} )^{5} \)
good3 \( 1 + 2 p^{2} T^{4} - 14 T^{6} + 241 T^{8} - 152 T^{10} + 241 p^{2} T^{12} - 14 p^{4} T^{14} + 2 p^{8} T^{16} + p^{10} T^{20} \)
7 \( 1 - 16 T^{2} + 230 T^{4} - 2294 T^{6} + 18985 T^{8} - 145540 T^{10} + 18985 p^{2} T^{12} - 2294 p^{4} T^{14} + 230 p^{6} T^{16} - 16 p^{8} T^{18} + p^{10} T^{20} \)
11 \( ( 1 - 8 T + 35 T^{2} - 130 T^{3} + 622 T^{4} - 2468 T^{5} + 622 p T^{6} - 130 p^{2} T^{7} + 35 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
17 \( 1 - 104 T^{2} + 5230 T^{4} - 171290 T^{6} + 4142945 T^{8} - 78706652 T^{10} + 4142945 p^{2} T^{12} - 171290 p^{4} T^{14} + 5230 p^{6} T^{16} - 104 p^{8} T^{18} + p^{10} T^{20} \)
19 \( ( 1 + 18 T + 9 p T^{2} + 1050 T^{3} + 4998 T^{4} + 21536 T^{5} + 4998 p T^{6} + 1050 p^{2} T^{7} + 9 p^{4} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
23 \( 1 - 90 T^{2} + 4497 T^{4} - 154332 T^{6} + 4288014 T^{8} - 103285236 T^{10} + 4288014 p^{2} T^{12} - 154332 p^{4} T^{14} + 4497 p^{6} T^{16} - 90 p^{8} T^{18} + p^{10} T^{20} \)
29 \( ( 1 + 2 T + 17 T^{2} + 36 T^{3} + 714 T^{4} + 7956 T^{5} + 714 p T^{6} + 36 p^{2} T^{7} + 17 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
31 \( ( 1 + 4 T + 83 T^{2} + 238 T^{3} + 3474 T^{4} + 9012 T^{5} + 3474 p T^{6} + 238 p^{2} T^{7} + 83 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
37 \( 1 - 176 T^{2} + 17358 T^{4} - 1211702 T^{6} + 64136241 T^{8} - 2663138948 T^{10} + 64136241 p^{2} T^{12} - 1211702 p^{4} T^{14} + 17358 p^{6} T^{16} - 176 p^{8} T^{18} + p^{10} T^{20} \)
41 \( ( 1 + 12 T + 157 T^{2} + 1424 T^{3} + 11674 T^{4} + 81544 T^{5} + 11674 p T^{6} + 1424 p^{2} T^{7} + 157 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
43 \( 1 - 252 T^{2} + 33714 T^{4} - 2995578 T^{6} + 194337801 T^{8} - 9542242520 T^{10} + 194337801 p^{2} T^{12} - 2995578 p^{4} T^{14} + 33714 p^{6} T^{16} - 252 p^{8} T^{18} + p^{10} T^{20} \)
47 \( 1 - 336 T^{2} + 54022 T^{4} - 5537670 T^{6} + 403256937 T^{8} - 21837719012 T^{10} + 403256937 p^{2} T^{12} - 5537670 p^{4} T^{14} + 54022 p^{6} T^{16} - 336 p^{8} T^{18} + p^{10} T^{20} \)
53 \( 1 - 290 T^{2} + 40133 T^{4} - 3609560 T^{6} + 245867826 T^{8} - 13991172300 T^{10} + 245867826 p^{2} T^{12} - 3609560 p^{4} T^{14} + 40133 p^{6} T^{16} - 290 p^{8} T^{18} + p^{10} T^{20} \)
59 \( ( 1 + 14 T + 287 T^{2} + 2574 T^{3} + 30242 T^{4} + 202256 T^{5} + 30242 p T^{6} + 2574 p^{2} T^{7} + 287 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
61 \( ( 1 - 10 T + 97 T^{2} - 172 T^{3} - 4630 T^{4} + 54908 T^{5} - 4630 p T^{6} - 172 p^{2} T^{7} + 97 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
67 \( 1 - 306 T^{2} + 58133 T^{4} - 7473864 T^{6} + 734030386 T^{8} - 55278015436 T^{10} + 734030386 p^{2} T^{12} - 7473864 p^{4} T^{14} + 58133 p^{6} T^{16} - 306 p^{8} T^{18} + p^{10} T^{20} \)
71 \( ( 1 - 18 T + 292 T^{2} - 2548 T^{3} + 26251 T^{4} - 186110 T^{5} + 26251 p T^{6} - 2548 p^{2} T^{7} + 292 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
73 \( 1 - 378 T^{2} + 74813 T^{4} - 10254120 T^{6} + 1070653906 T^{8} - 87825960124 T^{10} + 1070653906 p^{2} T^{12} - 10254120 p^{4} T^{14} + 74813 p^{6} T^{16} - 378 p^{8} T^{18} + p^{10} T^{20} \)
79 \( ( 1 + 8 T + 283 T^{2} + 1712 T^{3} + 38426 T^{4} + 185360 T^{5} + 38426 p T^{6} + 1712 p^{2} T^{7} + 283 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
83 \( 1 - 258 T^{2} + 36533 T^{4} - 4489800 T^{6} + 482931826 T^{8} - 43292500524 T^{10} + 482931826 p^{2} T^{12} - 4489800 p^{4} T^{14} + 36533 p^{6} T^{16} - 258 p^{8} T^{18} + p^{10} T^{20} \)
89 \( ( 1 - 6 T + 285 T^{2} - 984 T^{3} + 36986 T^{4} - 82308 T^{5} + 36986 p T^{6} - 984 p^{2} T^{7} + 285 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
97 \( 1 - 646 T^{2} + 196077 T^{4} - 37932360 T^{6} + 5334535122 T^{8} - 581791854692 T^{10} + 5334535122 p^{2} T^{12} - 37932360 p^{4} T^{14} + 196077 p^{6} T^{16} - 646 p^{8} T^{18} + p^{10} T^{20} \)
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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

−4.04591301687035372064807842114, −3.92618849234627973025502420122, −3.84931034237423181685216741008, −3.82193953122671489114638751039, −3.69669896238922479575889006065, −3.62926577332888476764766584018, −3.50326100497040140227899327500, −3.47672660338380071436395554341, −3.17890192307839114023878828722, −2.95372565690340562798611614995, −2.91711355419428892569131487728, −2.67601269624644218940535984285, −2.40666007709951260300290622231, −2.35106010345658752839802760093, −2.34229342734450385397590730405, −2.05135698512435975510062182462, −1.96481019865096816846261115425, −1.74056703651312963783787022176, −1.62422947273143985885898424055, −1.54633977486016815100548592573, −1.33863381248661066944391940406, −1.24771495739901540511014618257, −0.56412106327779268722117982568, −0.52188902491342571500790953199, −0.22525753530784877543359656574, 0.22525753530784877543359656574, 0.52188902491342571500790953199, 0.56412106327779268722117982568, 1.24771495739901540511014618257, 1.33863381248661066944391940406, 1.54633977486016815100548592573, 1.62422947273143985885898424055, 1.74056703651312963783787022176, 1.96481019865096816846261115425, 2.05135698512435975510062182462, 2.34229342734450385397590730405, 2.35106010345658752839802760093, 2.40666007709951260300290622231, 2.67601269624644218940535984285, 2.91711355419428892569131487728, 2.95372565690340562798611614995, 3.17890192307839114023878828722, 3.47672660338380071436395554341, 3.50326100497040140227899327500, 3.62926577332888476764766584018, 3.69669896238922479575889006065, 3.82193953122671489114638751039, 3.84931034237423181685216741008, 3.92618849234627973025502420122, 4.04591301687035372064807842114

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

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