↑

Properties

Label 16-546e8-1.1-c1e8-0-7
Degree $16$
Conductor $7.898\times 10^{21}$
Sign $1$
Analytic cond. $130544.$
Root an. cond. $2.08802$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s + 4·9-s + 8·11-s − 4·13-s − 2·16-s − 20·17-s − 20·19-s + 12·23-s + 8·25-s + 16·31-s − 24·37-s + 20·41-s + 16·45-s + 32·55-s + 20·61-s − 16·65-s + 24·67-s − 28·71-s + 16·73-s + 24·79-s − 8·80-s − 6·81-s + 28·83-s − 80·85-s − 80·95-s + 32·97-s + 32·99-s + ⋯
L(s)  = 1  + 1.78·5-s + 4/3·9-s + 2.41·11-s − 1.10·13-s − 1/2·16-s − 4.85·17-s − 4.58·19-s + 2.50·23-s + 8/5·25-s + 2.87·31-s − 3.94·37-s + 3.12·41-s + 2.38·45-s + 4.31·55-s + 2.56·61-s − 1.98·65-s + 2.93·67-s − 3.32·71-s + 1.87·73-s + 2.70·79-s − 0.894·80-s − 2/3·81-s + 3.07·83-s − 8.67·85-s − 8.20·95-s + 3.24·97-s + 3.21·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(130544.\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.341371458\)
\(L(\frac12)\) \(\approx\) \(5.341371458\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{4} )^{2} \)
3 \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
7 \( ( 1 + T^{4} )^{2} \)
13 \( 1 + 4 T + 22 T^{2} + 116 T^{3} + 418 T^{4} + 116 p T^{5} + 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
good5 \( 1 - 4 T + 8 T^{2} - 8 T^{3} - 39 T^{4} + 128 T^{5} - 168 T^{6} - 84 T^{7} + 1024 T^{8} - 84 p T^{9} - 168 p^{2} T^{10} + 128 p^{3} T^{11} - 39 p^{4} T^{12} - 8 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
11 \( 1 - 8 T + 32 T^{2} - 136 T^{3} + 27 p T^{4} + 448 T^{5} - 3840 T^{6} + 21968 T^{7} - 100480 T^{8} + 21968 p T^{9} - 3840 p^{2} T^{10} + 448 p^{3} T^{11} + 27 p^{5} T^{12} - 136 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
17 \( ( 1 + 10 T + 87 T^{2} + 466 T^{3} + 8 p^{2} T^{4} + 466 p T^{5} + 87 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( 1 + 20 T + 200 T^{2} + 1464 T^{3} + 9137 T^{4} + 50824 T^{5} + 260728 T^{6} + 1268724 T^{7} + 5773360 T^{8} + 1268724 p T^{9} + 260728 p^{2} T^{10} + 50824 p^{3} T^{11} + 9137 p^{4} T^{12} + 1464 p^{5} T^{13} + 200 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \)
23 \( ( 1 - 6 T + 3 T^{2} - 106 T^{3} + 58 p T^{4} - 106 p T^{5} + 3 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( 1 - 182 T^{2} + 15565 T^{4} - 815514 T^{6} + 28584632 T^{8} - 815514 p^{2} T^{10} + 15565 p^{4} T^{12} - 182 p^{6} T^{14} + p^{8} T^{16} \)
31 \( ( 1 - 8 T + 32 T^{2} - 280 T^{3} + 2434 T^{4} - 280 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 + 24 T + 288 T^{2} + 2720 T^{3} + 24641 T^{4} + 203216 T^{5} + 1479776 T^{6} + 10041848 T^{7} + 63855504 T^{8} + 10041848 p T^{9} + 1479776 p^{2} T^{10} + 203216 p^{3} T^{11} + 24641 p^{4} T^{12} + 2720 p^{5} T^{13} + 288 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 - 20 T + 200 T^{2} - 1580 T^{3} + 12160 T^{4} - 91380 T^{5} + 643800 T^{6} - 4270220 T^{7} + 27564222 T^{8} - 4270220 p T^{9} + 643800 p^{2} T^{10} - 91380 p^{3} T^{11} + 12160 p^{4} T^{12} - 1580 p^{5} T^{13} + 200 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 - 114 T^{2} + 9777 T^{4} - 576330 T^{6} + 27967892 T^{8} - 576330 p^{2} T^{10} + 9777 p^{4} T^{12} - 114 p^{6} T^{14} + p^{8} T^{16} \)
47 \( ( 1 - 1054 T^{4} + p^{4} T^{8} )^{2} \)
53 \( 1 - 20 T^{2} + 2648 T^{4} - 74940 T^{6} - 566130 T^{8} - 74940 p^{2} T^{10} + 2648 p^{4} T^{12} - 20 p^{6} T^{14} + p^{8} T^{16} \)
59 \( 1 - 448 T^{3} - 4676 T^{4} + 12992 T^{5} + 100352 T^{6} + 456064 T^{7} + 8885094 T^{8} + 456064 p T^{9} + 100352 p^{2} T^{10} + 12992 p^{3} T^{11} - 4676 p^{4} T^{12} - 448 p^{5} T^{13} + p^{8} T^{16} \)
61 \( ( 1 - 10 T + 43 T^{2} + 630 T^{3} - 6934 T^{4} + 630 p T^{5} + 43 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 24 T + 288 T^{2} - 3112 T^{3} + 22220 T^{4} - 22792 T^{5} - 1010080 T^{6} + 18275336 T^{7} - 204936186 T^{8} + 18275336 p T^{9} - 1010080 p^{2} T^{10} - 22792 p^{3} T^{11} + 22220 p^{4} T^{12} - 3112 p^{5} T^{13} + 288 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 + 28 T + 392 T^{2} + 4956 T^{3} + 64160 T^{4} + 687484 T^{5} + 6379800 T^{6} + 61128060 T^{7} + 559443454 T^{8} + 61128060 p T^{9} + 6379800 p^{2} T^{10} + 687484 p^{3} T^{11} + 64160 p^{4} T^{12} + 4956 p^{5} T^{13} + 392 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 - 16 T + 128 T^{2} - 1848 T^{3} + 15185 T^{4} - 68048 T^{5} + 160 p^{2} T^{6} - 74904 p T^{7} + 10278688 T^{8} - 74904 p^{2} T^{9} + 160 p^{4} T^{10} - 68048 p^{3} T^{11} + 15185 p^{4} T^{12} - 1848 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
79 \( ( 1 - 12 T + 326 T^{2} - 2604 T^{3} + 38530 T^{4} - 2604 p T^{5} + 326 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 28 T + 392 T^{2} - 5292 T^{3} + 76832 T^{4} - 883708 T^{5} + 8628312 T^{6} - 90708492 T^{7} + 915650206 T^{8} - 90708492 p T^{9} + 8628312 p^{2} T^{10} - 883708 p^{3} T^{11} + 76832 p^{4} T^{12} - 5292 p^{5} T^{13} + 392 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 - 1792 T^{3} + 11164 T^{4} - 1792 T^{5} + 1605632 T^{6} - 17099264 T^{7} + 5728134 T^{8} - 17099264 p T^{9} + 1605632 p^{2} T^{10} - 1792 p^{3} T^{11} + 11164 p^{4} T^{12} - 1792 p^{5} T^{13} + p^{8} T^{16} \)
97 \( ( 1 - 16 T + 128 T^{2} - 2000 T^{3} + 30466 T^{4} - 2000 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.76438805576133593997273516759, −4.67281191996678469079546944825, −4.51627957583937914779704384421, −4.37159266007857933268579128556, −4.22447993155692536548512845121, −3.97056469357228019911659051120, −3.94446731388682047110086081258, −3.91804197610050132852614031031, −3.80072261197272969321907442285, −3.42469342065461711885922529472, −3.29787920540181946892354421159, −3.02771712783623015605989258346, −2.95015511592598679774404067763, −2.54251773902658892310528236348, −2.52290677397076632193115676772, −2.23029498828409524118879313850, −2.11170356347331659034148010557, −2.10559217804106949163945249044, −1.98757290243378466457182204293, −1.95524112417897518011939293348, −1.73475944191955665161459484331, −1.20221241649692057442109946936, −0.884753383857394514679590650690, −0.812501359471237549407321044193, −0.35679254792895377623444869825, 0.35679254792895377623444869825, 0.812501359471237549407321044193, 0.884753383857394514679590650690, 1.20221241649692057442109946936, 1.73475944191955665161459484331, 1.95524112417897518011939293348, 1.98757290243378466457182204293, 2.10559217804106949163945249044, 2.11170356347331659034148010557, 2.23029498828409524118879313850, 2.52290677397076632193115676772, 2.54251773902658892310528236348, 2.95015511592598679774404067763, 3.02771712783623015605989258346, 3.29787920540181946892354421159, 3.42469342065461711885922529472, 3.80072261197272969321907442285, 3.91804197610050132852614031031, 3.94446731388682047110086081258, 3.97056469357228019911659051120, 4.22447993155692536548512845121, 4.37159266007857933268579128556, 4.51627957583937914779704384421, 4.67281191996678469079546944825, 4.76438805576133593997273516759

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

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