Properties

Label 16-42e16-1.1-c2e8-0-2
Degree $16$
Conductor $9.375\times 10^{25}$
Sign $1$
Analytic cond. $2.84884\times 10^{13}$
Root an. cond. $6.93293$
Motivic weight $2$
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  − 8·25-s + 128·37-s − 80·43-s − 384·67-s − 560·79-s − 400·109-s + 624·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.28e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  − 0.319·25-s + 3.45·37-s − 1.86·43-s − 5.73·67-s − 7.08·79-s − 3.66·109-s + 5.15·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 7.57·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(2.84884\times 10^{13}\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6505724650\)
\(L(\frac12)\) \(\approx\) \(0.6505724650\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( ( 1 + 4 T^{2} - 294 T^{4} + 4 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
11 \( ( 1 - 156 T^{2} + p^{4} T^{4} )^{4} \)
13 \( ( 1 + 320 T^{2} + p^{4} T^{4} )^{4} \)
17 \( ( 1 - 908 T^{2} + 359226 T^{4} - 908 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
19 \( ( 1 - 204 T^{2} + 116246 T^{4} - 204 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
23 \( ( 1 - 1800 T^{2} + 1344914 T^{4} - 1800 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
29 \( ( 1 - 2616 T^{2} + 3026354 T^{4} - 2616 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
31 \( ( 1 + 2260 T^{2} + 3068214 T^{4} + 2260 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
37 \( ( 1 - 32 T + 1446 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
41 \( ( 1 - 2588 T^{2} + 6977658 T^{4} - 2588 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
43 \( ( 1 + 20 T + 2250 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
47 \( ( 1 - 3932 T^{2} + 8769990 T^{4} - 3932 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
53 \( ( 1 - 5568 T^{2} + 15506786 T^{4} - 5568 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
59 \( ( 1 - 9020 T^{2} + 39720294 T^{4} - 9020 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
61 \( ( 1 + 12552 T^{2} + 65866226 T^{4} + 12552 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
67 \( ( 1 + 96 T + 9734 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
71 \( ( 1 - 9400 T^{2} + 11634 p^{2} T^{4} - 9400 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
73 \( ( 1 + 12024 T^{2} + 80952914 T^{4} + 12024 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
79 \( ( 1 + 140 T + 11190 T^{2} + 140 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
83 \( ( 1 - 6404 T^{2} + 98011494 T^{4} - 6404 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( ( 1 - 16988 T^{2} + 195278010 T^{4} - 16988 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
97 \( ( 1 + 13704 T^{2} + 84688466 T^{4} + 13704 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
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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

−3.70703467345353664418581278477, −3.62540107724576813383793334845, −3.62296508853656455002283753460, −3.32877641164977747199426581357, −3.21757506687676534129070232299, −2.89561378517913763723603586688, −2.84994237954167095424372908182, −2.81213703765693816030004712712, −2.67996748793577988485862282488, −2.57446025297785277859077389674, −2.57264405240249497580459224104, −2.29609296979220901801618879384, −2.26337729490059930721476044829, −2.07535012365434887037994773936, −1.69985729415175092847096371916, −1.49114715538849250800669231119, −1.37791767062607346548234643920, −1.30928061433240004302869737253, −1.30655400062952247127484198251, −1.30012447548869083030607302911, −1.11363613402187220079407239634, −0.56174532089995450249990047182, −0.39557202292036436888524309397, −0.17113582141345601652867629233, −0.12140160820408946687575332703, 0.12140160820408946687575332703, 0.17113582141345601652867629233, 0.39557202292036436888524309397, 0.56174532089995450249990047182, 1.11363613402187220079407239634, 1.30012447548869083030607302911, 1.30655400062952247127484198251, 1.30928061433240004302869737253, 1.37791767062607346548234643920, 1.49114715538849250800669231119, 1.69985729415175092847096371916, 2.07535012365434887037994773936, 2.26337729490059930721476044829, 2.29609296979220901801618879384, 2.57264405240249497580459224104, 2.57446025297785277859077389674, 2.67996748793577988485862282488, 2.81213703765693816030004712712, 2.84994237954167095424372908182, 2.89561378517913763723603586688, 3.21757506687676534129070232299, 3.32877641164977747199426581357, 3.62296508853656455002283753460, 3.62540107724576813383793334845, 3.70703467345353664418581278477

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

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