Properties

Label 16-384e8-1.1-c4e8-0-0
Degree $16$
Conductor $4.728\times 10^{20}$
Sign $1$
Analytic cond. $6.16317\times 10^{12}$
Root an. cond. $6.30032$
Motivic weight $4$
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  + 108·9-s + 1.39e3·17-s + 712·25-s − 1.00e3·41-s + 1.47e4·49-s − 1.02e4·73-s + 7.29e3·81-s − 3.16e4·89-s − 4.52e4·97-s − 240·113-s − 2.63e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.50e5·153-s + 157-s + 163-s + 167-s + 1.54e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 4/3·9-s + 4.81·17-s + 1.13·25-s − 0.599·41-s + 6.16·49-s − 1.92·73-s + 10/9·81-s − 3.99·89-s − 4.80·97-s − 0.0187·113-s − 1.79·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 6.42·153-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 5.41·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{56} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(6.16317\times 10^{12}\)
Root analytic conductor: \(6.30032\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{56} \cdot 3^{8} ,\ ( \ : [2]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(15.10546127\)
\(L(\frac12)\) \(\approx\) \(15.10546127\)
\(L(3)\) 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 \( ( 1 - p^{3} T^{2} )^{4} \)
good5 \( ( 1 - 356 T^{2} + 554886 T^{4} - 356 p^{8} T^{6} + p^{16} T^{8} )^{2} \)
7 \( ( 1 - 7396 T^{2} + 24172614 T^{4} - 7396 p^{8} T^{6} + p^{16} T^{8} )^{2} \)
11 \( ( 1 + 1196 p T^{2} + 368768646 T^{4} + 1196 p^{9} T^{6} + p^{16} T^{8} )^{2} \)
13 \( ( 1 - 77380 T^{2} + 2896134342 T^{4} - 77380 p^{8} T^{6} + p^{16} T^{8} )^{2} \)
17 \( ( 1 - 348 T + 132806 T^{2} - 348 p^{4} T^{3} + p^{8} T^{4} )^{4} \)
19 \( ( 1 + 29668 T^{2} - 6278881530 T^{4} + 29668 p^{8} T^{6} + p^{16} T^{8} )^{2} \)
23 \( ( 1 - 293380 T^{2} + 44367843462 T^{4} - 293380 p^{8} T^{6} + p^{16} T^{8} )^{2} \)
29 \( ( 1 - 1797092 T^{2} + 2530086 p^{4} T^{4} - 1797092 p^{8} T^{6} + p^{16} T^{8} )^{2} \)
31 \( ( 1 - 2304868 T^{2} + 2740651806150 T^{4} - 2304868 p^{8} T^{6} + p^{16} T^{8} )^{2} \)
37 \( ( 1 - 6633796 T^{2} + 17981251583046 T^{4} - 6633796 p^{8} T^{6} + p^{16} T^{8} )^{2} \)
41 \( ( 1 + 252 T + 441926 T^{2} + 252 p^{4} T^{3} + p^{8} T^{4} )^{4} \)
43 \( ( 1 + 8877412 T^{2} + 38500496392326 T^{4} + 8877412 p^{8} T^{6} + p^{16} T^{8} )^{2} \)
47 \( ( 1 - 2352772 T^{2} + 30062992642566 T^{4} - 2352772 p^{8} T^{6} + p^{16} T^{8} )^{2} \)
53 \( ( 1 - 30298724 T^{2} + 354018517829766 T^{4} - 30298724 p^{8} T^{6} + p^{16} T^{8} )^{2} \)
59 \( ( 1 + 22520114 T^{2} + p^{8} T^{4} )^{4} \)
61 \( ( 1 - 31835332 T^{2} + 512888648414406 T^{4} - 31835332 p^{8} T^{6} + p^{16} T^{8} )^{2} \)
67 \( ( 1 + 64146148 T^{2} + 1777243586102790 T^{4} + 64146148 p^{8} T^{6} + p^{16} T^{8} )^{2} \)
71 \( ( 1 - 6928132 T^{2} - 515199567726714 T^{4} - 6928132 p^{8} T^{6} + p^{16} T^{8} )^{2} \)
73 \( ( 1 + 2564 T + 57407814 T^{2} + 2564 p^{4} T^{3} + p^{8} T^{4} )^{4} \)
79 \( ( 1 - 8064100 T^{2} + 1391113872282822 T^{4} - 8064100 p^{8} T^{6} + p^{16} T^{8} )^{2} \)
83 \( ( 1 + 127863268 T^{2} + 8102648542271238 T^{4} + 127863268 p^{8} T^{6} + p^{16} T^{8} )^{2} \)
89 \( ( 1 + 7908 T + 134667398 T^{2} + 7908 p^{4} T^{3} + p^{8} T^{4} )^{4} \)
97 \( ( 1 + 11300 T + 202529862 T^{2} + 11300 p^{4} T^{3} + p^{8} T^{4} )^{4} \)
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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

−4.27581227997440029755409440735, −4.24973012910214436680436429008, −4.06129343924423344890375528613, −3.74909956355274498553773573873, −3.69302260942821602468342415591, −3.65602649165102490158983450169, −3.44811617809558209244076337450, −3.30773212524639191830132547631, −3.10232688147148742536401163201, −2.88154398004395523842857332563, −2.80475059434199333864809438717, −2.56927416791414432896579464985, −2.51713201098634714734369018601, −2.42271228042856419361943939839, −2.03506882806073001920297222615, −1.88129499231358913351950896578, −1.45610442659753952856946419237, −1.40402525523466349098537537724, −1.31084055014750739853321333572, −1.20375361168000848898650642868, −1.03795742359696888292648267858, −0.858276197064239112928949940639, −0.66433513914027309151302908042, −0.33753040508767507228070569857, −0.20660057759175038648122353339, 0.20660057759175038648122353339, 0.33753040508767507228070569857, 0.66433513914027309151302908042, 0.858276197064239112928949940639, 1.03795742359696888292648267858, 1.20375361168000848898650642868, 1.31084055014750739853321333572, 1.40402525523466349098537537724, 1.45610442659753952856946419237, 1.88129499231358913351950896578, 2.03506882806073001920297222615, 2.42271228042856419361943939839, 2.51713201098634714734369018601, 2.56927416791414432896579464985, 2.80475059434199333864809438717, 2.88154398004395523842857332563, 3.10232688147148742536401163201, 3.30773212524639191830132547631, 3.44811617809558209244076337450, 3.65602649165102490158983450169, 3.69302260942821602468342415591, 3.74909956355274498553773573873, 4.06129343924423344890375528613, 4.24973012910214436680436429008, 4.27581227997440029755409440735

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

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