Properties

Label 16-825e8-1.1-c1e8-0-11
Degree $16$
Conductor $2.146\times 10^{23}$
Sign $1$
Analytic cond. $3.54689\times 10^{6}$
Root an. cond. $2.56664$
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  + 9-s + 8·11-s − 6·16-s + 10·19-s + 10·29-s − 14·31-s − 34·41-s − 30·49-s + 30·59-s − 14·61-s − 5·64-s + 16·71-s − 30·79-s − 60·89-s + 8·99-s + 26·101-s − 60·109-s + 36·121-s + 127-s + 131-s + 137-s + 139-s − 6·144-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 1/3·9-s + 2.41·11-s − 3/2·16-s + 2.29·19-s + 1.85·29-s − 2.51·31-s − 5.30·41-s − 4.28·49-s + 3.90·59-s − 1.79·61-s − 5/8·64-s + 1.89·71-s − 3.37·79-s − 6.35·89-s + 0.804·99-s + 2.58·101-s − 5.74·109-s + 3.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 5^{16} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(3.54689\times 10^{6}\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 5^{16} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.500716080\)
\(L(\frac12)\) \(\approx\) \(2.500716080\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
5 \( 1 \)
11 \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
good2 \( 1 + 3 p T^{4} + 5 T^{6} + 21 T^{8} + 5 p^{2} T^{10} + 3 p^{5} T^{12} + p^{8} T^{16} \)
7 \( ( 1 - 2 T + 17 T^{2} - 10 T^{3} + 121 T^{4} - 10 p T^{5} + 17 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )( 1 + 2 T + 17 T^{2} + 10 T^{3} + 121 T^{4} + 10 p T^{5} + 17 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} ) \)
13 \( 1 + 10 T^{2} + 291 T^{4} + 4640 T^{6} + 41141 T^{8} + 4640 p^{2} T^{10} + 291 p^{4} T^{12} + 10 p^{6} T^{14} + p^{8} T^{16} \)
17 \( ( 1 - 8 T + 47 T^{2} - 240 T^{3} + 1121 T^{4} - 240 p T^{5} + 47 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )( 1 + 8 T + 47 T^{2} + 240 T^{3} + 1121 T^{4} + 240 p T^{5} + 47 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} ) \)
19 \( ( 1 - 5 T + 6 T^{2} + 65 T^{3} - 439 T^{4} + 65 p T^{5} + 6 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( ( 1 - 65 T^{2} + 2053 T^{4} - 65 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 5 T + 11 T^{2} - 195 T^{3} + 1736 T^{4} - 195 p T^{5} + 11 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 7 T + 18 T^{2} - 91 T^{3} - 1195 T^{4} - 91 p T^{5} + 18 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 15 T^{2} + 2816 T^{4} + 13815 T^{6} + 3638791 T^{8} + 13815 p^{2} T^{10} + 2816 p^{4} T^{12} - 15 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 + 17 T + 68 T^{2} - 661 T^{3} - 8425 T^{4} - 661 p T^{5} + 68 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 85 T^{2} + 3973 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( 1 + 75 T^{2} + 1791 T^{4} - 82375 T^{6} - 9373944 T^{8} - 82375 p^{2} T^{10} + 1791 p^{4} T^{12} + 75 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 + 95 T^{2} + 1731 T^{4} - 240275 T^{6} - 22192624 T^{8} - 240275 p^{2} T^{10} + 1731 p^{4} T^{12} + 95 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 - 15 T + 76 T^{2} - 675 T^{3} + 8161 T^{4} - 675 p T^{5} + 76 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 7 T - 12 T^{2} - 511 T^{3} - 2845 T^{4} - 511 p T^{5} - 12 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( ( 1 - 45 T^{2} + 8933 T^{4} - 45 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - 8 T - 7 T^{2} + 624 T^{3} - 4495 T^{4} + 624 p T^{5} - 7 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 - 135 T^{2} + 16136 T^{4} - 1481985 T^{6} + 136733311 T^{8} - 1481985 p^{2} T^{10} + 16136 p^{4} T^{12} - 135 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 + 15 T + 21 T^{2} - 145 T^{3} + 2916 T^{4} - 145 p T^{5} + 21 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 + 90 T^{2} + 97 p T^{4} + 479640 T^{6} + 2902741 T^{8} + 479640 p^{2} T^{10} + 97 p^{5} T^{12} + 90 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 15 T + 223 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{4} \)
97 \( 1 + 265 T^{2} + 17976 T^{4} - 2527825 T^{6} - 507942289 T^{8} - 2527825 p^{2} T^{10} + 17976 p^{4} T^{12} + 265 p^{6} T^{14} + p^{8} T^{16} \)
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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

−4.48693901156370274588238865767, −4.15369942196862917289729335932, −4.12173024466254819935781746698, −4.01884336748027893554911683521, −3.94032804511189996035801695541, −3.82458476134419301968501853750, −3.57583232983817123478694661696, −3.29779580108506983368580819429, −3.28392685147656144055755253851, −3.23586211603942235530476408559, −3.19780678861399921008338114100, −2.85231998013456925733880370615, −2.80338769844465077623753462629, −2.75221317055096816481366266133, −2.38571547111697959282256902134, −2.15479521788513146443288003367, −1.85942904767549561768513675196, −1.69108236050849943620575193896, −1.69007076768921078179694390914, −1.65291311969125213208295169379, −1.32564395579228483047443811847, −1.29133026980669130427797980558, −0.941141408091541424160845685062, −0.44216532151717620211852097535, −0.24614996010214784158407187670, 0.24614996010214784158407187670, 0.44216532151717620211852097535, 0.941141408091541424160845685062, 1.29133026980669130427797980558, 1.32564395579228483047443811847, 1.65291311969125213208295169379, 1.69007076768921078179694390914, 1.69108236050849943620575193896, 1.85942904767549561768513675196, 2.15479521788513146443288003367, 2.38571547111697959282256902134, 2.75221317055096816481366266133, 2.80338769844465077623753462629, 2.85231998013456925733880370615, 3.19780678861399921008338114100, 3.23586211603942235530476408559, 3.28392685147656144055755253851, 3.29779580108506983368580819429, 3.57583232983817123478694661696, 3.82458476134419301968501853750, 3.94032804511189996035801695541, 4.01884336748027893554911683521, 4.12173024466254819935781746698, 4.15369942196862917289729335932, 4.48693901156370274588238865767

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

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