Properties

Label 16-560e8-1.1-c1e8-0-0
Degree $16$
Conductor $9.672\times 10^{21}$
Sign $1$
Analytic cond. $159853.$
Root an. cond. $2.11462$
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  − 6·9-s − 4·25-s − 4·29-s − 24·37-s − 10·49-s − 48·53-s + 23·81-s + 20·109-s − 56·113-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 30·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 2·9-s − 4/5·25-s − 0.742·29-s − 3.94·37-s − 1.42·49-s − 6.59·53-s + 23/9·81-s + 1.91·109-s − 5.26·113-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(159853.\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1690692013\)
\(L(\frac12)\) \(\approx\) \(0.1690692013\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 + T^{2} )^{4} \)
7 \( 1 + 10 T^{2} + 50 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \)
good3 \( ( 1 + p T^{2} + 2 T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 - 9 T^{2} + 244 T^{4} - 9 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - 15 T^{2} + 376 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( ( 1 - 31 T^{2} + 800 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 + 430 T^{4} + p^{4} T^{8} )^{2} \)
23 \( ( 1 - 54 T^{2} + 1714 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + T + 40 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \)
31 \( ( 1 + 88 T^{2} + 3566 T^{4} + 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 6 T + 10 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \)
41 \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{4} \)
43 \( ( 1 - 10 T^{2} - 2190 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 123 T^{2} + 7306 T^{4} + 123 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 + 6 T + p T^{2} )^{8} \)
59 \( ( 1 + 96 T^{2} + 8974 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 48 T^{2} + 718 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( ( 1 - 74 T^{2} + 6770 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - 144 T^{2} + 14974 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 16 T + p T^{2} )^{4}( 1 + 16 T + p T^{2} )^{4} \)
79 \( ( 1 - 145 T^{2} + 16260 T^{4} - 145 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 170 T^{2} + 15090 T^{4} + 170 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \)
97 \( ( 1 - 39 T^{2} + 7792 T^{4} - 39 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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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

−4.75575758033386263790945343447, −4.68495604607620072766661243269, −4.66652604355479149200925783203, −4.38770484604476266811966268300, −3.94823962588377243866904587265, −3.91574118070952209484185765662, −3.76848668091298248677010739802, −3.69384081864308584311199047856, −3.62193827634436889182504672196, −3.51611998449194819424859069563, −3.11645135941585900701702931226, −3.06080708187021258242744351199, −2.90549052726397714113491311142, −2.89307647508808994445479466494, −2.81400526152161706887455515732, −2.50683503058371447848529515310, −2.22835973262048444909466296013, −1.92406909586793803714898112729, −1.77872985195593248326968872668, −1.63473736122001056459159440224, −1.58799908677812452291160875365, −1.56437852005796828201709667723, −0.842786676772810231548423069779, −0.44564785480583072287505071437, −0.10347767445057255242964436078, 0.10347767445057255242964436078, 0.44564785480583072287505071437, 0.842786676772810231548423069779, 1.56437852005796828201709667723, 1.58799908677812452291160875365, 1.63473736122001056459159440224, 1.77872985195593248326968872668, 1.92406909586793803714898112729, 2.22835973262048444909466296013, 2.50683503058371447848529515310, 2.81400526152161706887455515732, 2.89307647508808994445479466494, 2.90549052726397714113491311142, 3.06080708187021258242744351199, 3.11645135941585900701702931226, 3.51611998449194819424859069563, 3.62193827634436889182504672196, 3.69384081864308584311199047856, 3.76848668091298248677010739802, 3.91574118070952209484185765662, 3.94823962588377243866904587265, 4.38770484604476266811966268300, 4.66652604355479149200925783203, 4.68495604607620072766661243269, 4.75575758033386263790945343447

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

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