Properties

Label 20-4600e10-1.1-c1e10-0-1
Degree $20$
Conductor $4.242\times 10^{36}$
Sign $1$
Analytic cond. $4.47043\times 10^{15}$
Root an. cond. $6.06062$
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  + 12·9-s − 24·29-s − 36·31-s − 12·41-s + 20·49-s + 2·59-s + 20·61-s + 16·71-s + 54·81-s − 28·89-s − 6·101-s − 84·109-s − 80·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 50·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 4·9-s − 4.45·29-s − 6.46·31-s − 1.87·41-s + 20/7·49-s + 0.260·59-s + 2.56·61-s + 1.89·71-s + 6·81-s − 2.96·89-s − 0.597·101-s − 8.04·109-s − 7.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 5^{20} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 5^{20} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{30} \cdot 5^{20} \cdot 23^{10}\)
Sign: $1$
Analytic conductor: \(4.47043\times 10^{15}\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{30} \cdot 5^{20} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.939983708\)
\(L(\frac12)\) \(\approx\) \(1.939983708\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23 \( ( 1 + T^{2} )^{5} \)
good3 \( 1 - 4 p T^{2} + 10 p^{2} T^{4} - 53 p^{2} T^{6} + 221 p^{2} T^{8} - 734 p^{2} T^{10} + 221 p^{4} T^{12} - 53 p^{6} T^{14} + 10 p^{8} T^{16} - 4 p^{9} T^{18} + p^{10} T^{20} \)
7 \( 1 - 20 T^{2} + 342 T^{4} - 3858 T^{6} + 5391 p T^{8} - 282420 T^{10} + 5391 p^{3} T^{12} - 3858 p^{4} T^{14} + 342 p^{6} T^{16} - 20 p^{8} T^{18} + p^{10} T^{20} \)
11 \( ( 1 + 40 T^{2} - 6 T^{3} + 763 T^{4} - 108 T^{5} + 763 p T^{6} - 6 p^{2} T^{7} + 40 p^{3} T^{8} + p^{5} T^{10} )^{2} \)
13 \( 1 - 50 T^{2} + 1611 T^{4} - 37083 T^{6} + 667083 T^{8} - 9654915 T^{10} + 667083 p^{2} T^{12} - 37083 p^{4} T^{14} + 1611 p^{6} T^{16} - 50 p^{8} T^{18} + p^{10} T^{20} \)
17 \( 1 - 38 T^{2} + 333 T^{4} + 504 T^{6} + 173778 T^{8} - 5816484 T^{10} + 173778 p^{2} T^{12} + 504 p^{4} T^{14} + 333 p^{6} T^{16} - 38 p^{8} T^{18} + p^{10} T^{20} \)
19 \( ( 1 + 80 T^{2} + 6 T^{3} + 2803 T^{4} + 204 T^{5} + 2803 p T^{6} + 6 p^{2} T^{7} + 80 p^{3} T^{8} + p^{5} T^{10} )^{2} \)
29 \( ( 1 + 12 T + 5 p T^{2} + 1053 T^{3} + 8137 T^{4} + 43677 T^{5} + 8137 p T^{6} + 1053 p^{2} T^{7} + 5 p^{4} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
31 \( ( 1 + 18 T + 8 p T^{2} + 2277 T^{3} + 17743 T^{4} + 106350 T^{5} + 17743 p T^{6} + 2277 p^{2} T^{7} + 8 p^{4} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
37 \( 1 - 206 T^{2} + 22149 T^{4} - 1610664 T^{6} + 87043026 T^{8} - 3641176404 T^{10} + 87043026 p^{2} T^{12} - 1610664 p^{4} T^{14} + 22149 p^{6} T^{16} - 206 p^{8} T^{18} + p^{10} T^{20} \)
41 \( ( 1 + 6 T + 139 T^{2} + 747 T^{3} + 10003 T^{4} + 41535 T^{5} + 10003 p T^{6} + 747 p^{2} T^{7} + 139 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
43 \( 1 - 284 T^{2} + 40326 T^{4} - 3728946 T^{6} + 247172361 T^{8} - 12229550436 T^{10} + 247172361 p^{2} T^{12} - 3728946 p^{4} T^{14} + 40326 p^{6} T^{16} - 284 p^{8} T^{18} + p^{10} T^{20} \)
47 \( 1 - 180 T^{2} + 12350 T^{4} - 566229 T^{6} + 38048065 T^{8} - 2341351326 T^{10} + 38048065 p^{2} T^{12} - 566229 p^{4} T^{14} + 12350 p^{6} T^{16} - 180 p^{8} T^{18} + p^{10} T^{20} \)
53 \( 1 - 246 T^{2} + 23861 T^{4} - 926952 T^{6} - 17707454 T^{8} + 3025689468 T^{10} - 17707454 p^{2} T^{12} - 926952 p^{4} T^{14} + 23861 p^{6} T^{16} - 246 p^{8} T^{18} + p^{10} T^{20} \)
59 \( ( 1 - T + 212 T^{2} - 343 T^{3} + 21441 T^{4} - 30508 T^{5} + 21441 p T^{6} - 343 p^{2} T^{7} + 212 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \)
61 \( ( 1 - 10 T + 105 T^{2} - 1368 T^{3} + 13410 T^{4} - 84828 T^{5} + 13410 p T^{6} - 1368 p^{2} T^{7} + 105 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
67 \( 1 - 230 T^{2} + 22629 T^{4} - 1115496 T^{6} + 16658082 T^{8} + 834707676 T^{10} + 16658082 p^{2} T^{12} - 1115496 p^{4} T^{14} + 22629 p^{6} T^{16} - 230 p^{8} T^{18} + p^{10} T^{20} \)
71 \( ( 1 - 8 T + 176 T^{2} - 1583 T^{3} + 18243 T^{4} - 136046 T^{5} + 18243 p T^{6} - 1583 p^{2} T^{7} + 176 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
73 \( 1 - 250 T^{2} + 30051 T^{4} - 1688787 T^{6} + 4039035 T^{8} + 4897858077 T^{10} + 4039035 p^{2} T^{12} - 1688787 p^{4} T^{14} + 30051 p^{6} T^{16} - 250 p^{8} T^{18} + p^{10} T^{20} \)
79 \( ( 1 + 242 T^{2} + 36 T^{3} + 31333 T^{4} + 7416 T^{5} + 31333 p T^{6} + 36 p^{2} T^{7} + 242 p^{3} T^{8} + p^{5} T^{10} )^{2} \)
83 \( 1 - 444 T^{2} + 95678 T^{4} - 13616394 T^{6} + 1469622337 T^{8} - 131494875156 T^{10} + 1469622337 p^{2} T^{12} - 13616394 p^{4} T^{14} + 95678 p^{6} T^{16} - 444 p^{8} T^{18} + p^{10} T^{20} \)
89 \( ( 1 + 14 T + 401 T^{2} + 3704 T^{3} + 62598 T^{4} + 434420 T^{5} + 62598 p T^{6} + 3704 p^{2} T^{7} + 401 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
97 \( 1 - 646 T^{2} + 203661 T^{4} - 41769480 T^{6} + 6180672306 T^{8} - 687578984676 T^{10} + 6180672306 p^{2} T^{12} - 41769480 p^{4} T^{14} + 203661 p^{6} T^{16} - 646 p^{8} T^{18} + p^{10} T^{20} \)
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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

−2.85835479148872131911889304559, −2.69208264437826901328369703911, −2.45599784625811697042424767458, −2.42980949879126577265849039031, −2.42267628275701367947052431690, −2.32245702870253218013955924472, −2.29668831547900638599572479695, −2.23086571519148205443368688112, −2.21733189626345870890880853643, −2.08001246605113863262942439326, −1.74025146257455691680137716623, −1.62136978557804193686578423208, −1.53747372939932583032889710649, −1.50795018660753883559115959854, −1.44781982575579223765552867465, −1.38683995820136882419082971015, −1.34168190487021342366926287444, −1.23794814030020725084723139214, −1.21194649752925823927698847749, −1.14823470356046305133745746349, −0.60401100749895926573771387613, −0.36652519075668368638498649860, −0.33643821475716445205698980173, −0.32639880268806939173996251056, −0.10334653937759163022429664367, 0.10334653937759163022429664367, 0.32639880268806939173996251056, 0.33643821475716445205698980173, 0.36652519075668368638498649860, 0.60401100749895926573771387613, 1.14823470356046305133745746349, 1.21194649752925823927698847749, 1.23794814030020725084723139214, 1.34168190487021342366926287444, 1.38683995820136882419082971015, 1.44781982575579223765552867465, 1.50795018660753883559115959854, 1.53747372939932583032889710649, 1.62136978557804193686578423208, 1.74025146257455691680137716623, 2.08001246605113863262942439326, 2.21733189626345870890880853643, 2.23086571519148205443368688112, 2.29668831547900638599572479695, 2.32245702870253218013955924472, 2.42267628275701367947052431690, 2.42980949879126577265849039031, 2.45599784625811697042424767458, 2.69208264437826901328369703911, 2.85835479148872131911889304559

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

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