Properties

Label 16-570e8-1.1-c1e8-0-6
Degree $16$
Conductor $1.114\times 10^{22}$
Sign $1$
Analytic cond. $184168.$
Root an. cond. $2.13341$
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  + 2·4-s + 4·5-s + 2·9-s + 16·11-s + 16-s + 8·19-s + 8·20-s + 8·25-s − 8·29-s − 24·31-s + 4·36-s + 32·44-s + 8·45-s + 20·49-s + 64·55-s + 8·59-s + 12·61-s − 2·64-s + 24·71-s + 16·76-s + 28·79-s + 4·80-s + 81-s − 56·89-s + 32·95-s + 32·99-s + 16·100-s + ⋯
L(s)  = 1  + 4-s + 1.78·5-s + 2/3·9-s + 4.82·11-s + 1/4·16-s + 1.83·19-s + 1.78·20-s + 8/5·25-s − 1.48·29-s − 4.31·31-s + 2/3·36-s + 4.82·44-s + 1.19·45-s + 20/7·49-s + 8.62·55-s + 1.04·59-s + 1.53·61-s − 1/4·64-s + 2.84·71-s + 1.83·76-s + 3.15·79-s + 0.447·80-s + 1/9·81-s − 5.93·89-s + 3.28·95-s + 3.21·99-s + 8/5·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(184168.\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(18.19538666\)
\(L(\frac12)\) \(\approx\) \(18.19538666\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} + T^{4} )^{2} \)
3 \( ( 1 - T^{2} + T^{4} )^{2} \)
5 \( 1 - 4 T + 8 T^{2} + 8 T^{3} - 41 T^{4} + 8 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
19 \( ( 1 - 2 T + p T^{2} )^{4} \)
good7 \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{4} \)
11 \( ( 1 - 2 T + p T^{2} )^{8} \)
13 \( 1 + 38 T^{2} + 769 T^{4} + 12806 T^{6} + 183028 T^{8} + 12806 p^{2} T^{10} + 769 p^{4} T^{12} + 38 p^{6} T^{14} + p^{8} T^{16} \)
17 \( ( 1 + 10 T^{2} - 189 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 + 40 T^{2} + 1071 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + 4 T + 8 T^{2} - 200 T^{3} - 1241 T^{4} - 200 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 6 T + 47 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \)
37 \( ( 1 - 38 T^{2} + 1923 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
41 \( ( 1 - 76 T^{2} + 4095 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( 1 + 142 T^{2} + 11641 T^{4} + 685150 T^{6} + 31939492 T^{8} + 685150 p^{2} T^{10} + 11641 p^{4} T^{12} + 142 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 + 104 T^{2} + 4558 T^{4} + 191360 T^{6} + 10390339 T^{8} + 191360 p^{2} T^{10} + 4558 p^{4} T^{12} + 104 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 + 36 T^{2} + 1498 T^{4} - 209520 T^{6} - 11490141 T^{8} - 209520 p^{2} T^{10} + 1498 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 - 4 T - 10 T^{2} + 368 T^{3} - 3749 T^{4} + 368 p T^{5} - 10 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 6 T - 41 T^{2} + 270 T^{3} + 12 T^{4} + 270 p T^{5} - 41 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 + 254 T^{2} + 39433 T^{4} + 4090670 T^{6} + 319393444 T^{8} + 4090670 p^{2} T^{10} + 39433 p^{4} T^{12} + 254 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 - 12 T - 28 T^{2} - 360 T^{3} + 14319 T^{4} - 360 p T^{5} - 28 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 + 2 p T^{2} + 10033 T^{4} + 1250 p T^{6} - 14685116 T^{8} + 1250 p^{3} T^{10} + 10033 p^{4} T^{12} + 2 p^{7} T^{14} + p^{8} T^{16} \)
79 \( ( 1 - 14 T + 85 T^{2} + 658 T^{3} - 9404 T^{4} + 658 p T^{5} + 85 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( ( 1 - 152 T^{2} + 11778 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 28 T + 416 T^{2} + 5320 T^{3} + 57727 T^{4} + 5320 p T^{5} + 416 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 - 76 T^{2} - 662 T^{4} + 940880 T^{6} - 101427821 T^{8} + 940880 p^{2} T^{10} - 662 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.82392331952218219030606050288, −4.28231677495373254959296845418, −4.22836728562428842150314098739, −4.20140799789345866755404427896, −4.14612999618081273438881165490, −3.99783345304522377314605243596, −3.81582510883488983519931164169, −3.73134574320151464956804465359, −3.63420317726536867041668183230, −3.58112059221371697357359397729, −3.28370245451862360023301902396, −3.16515501129930089214983614345, −2.79481406502165989973807432770, −2.73389084361539852638982537517, −2.56330472061208517004571420226, −2.33821348194084421293385029238, −2.16568300575501707031925681257, −1.92281266071465925913366163613, −1.91163064807301219454627926060, −1.56037044057019675620847923533, −1.42250200333678320591460541703, −1.33252892601386570058668800920, −1.25678251670961016671837989348, −0.953929358926817826036942558078, −0.47116116087369537822635055725, 0.47116116087369537822635055725, 0.953929358926817826036942558078, 1.25678251670961016671837989348, 1.33252892601386570058668800920, 1.42250200333678320591460541703, 1.56037044057019675620847923533, 1.91163064807301219454627926060, 1.92281266071465925913366163613, 2.16568300575501707031925681257, 2.33821348194084421293385029238, 2.56330472061208517004571420226, 2.73389084361539852638982537517, 2.79481406502165989973807432770, 3.16515501129930089214983614345, 3.28370245451862360023301902396, 3.58112059221371697357359397729, 3.63420317726536867041668183230, 3.73134574320151464956804465359, 3.81582510883488983519931164169, 3.99783345304522377314605243596, 4.14612999618081273438881165490, 4.20140799789345866755404427896, 4.22836728562428842150314098739, 4.28231677495373254959296845418, 4.82392331952218219030606050288

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

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