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Properties

Label 16-3332e8-1.1-c0e8-0-8
Degree $16$
Conductor $1.519\times 10^{28}$
Sign $1$
Analytic cond. $58.4651$
Root an. cond. $1.28952$
Motivic weight $0$
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  + 8·13-s + 8·37-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯
L(s)  = 1  + 8·13-s + 8·37-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 7^{16} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(58.4651\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 7^{16} \cdot 17^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(7.736500008\)
\(L(\frac12)\) \(\approx\) \(7.736500008\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{8} \)
7 \( 1 \)
17 \( 1 + T^{8} \)
good3 \( 1 + T^{16} \)
5 \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \)
11 \( 1 + T^{16} \)
13 \( ( 1 - T )^{8}( 1 + T^{2} )^{4} \)
19 \( ( 1 + T^{8} )^{2} \)
23 \( 1 + T^{16} \)
29 \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \)
31 \( 1 + T^{16} \)
37 \( ( 1 - T )^{8}( 1 + T^{8} ) \)
41 \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \)
43 \( ( 1 + T^{8} )^{2} \)
47 \( ( 1 + T^{4} )^{4} \)
53 \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \)
59 \( ( 1 + T^{8} )^{2} \)
61 \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \)
67 \( ( 1 + T^{2} )^{8} \)
71 \( 1 + T^{16} \)
73 \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \)
79 \( 1 + T^{16} \)
83 \( ( 1 + T^{8} )^{2} \)
89 \( ( 1 + T^{4} )^{4} \)
97 \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \)
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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

−3.74525691457417818661471107468, −3.68834038120134269828626261613, −3.67912524995280702781805175386, −3.64220783691536117804935198785, −3.27088980362568678000930031419, −3.18872286981222964428811767547, −3.11125772861780942345976302348, −2.99750412133669500201882345690, −2.91933398413382598881907258715, −2.87679124674465301627324536735, −2.64396345573457361494595660662, −2.51016142409128919257421190712, −2.32067262049788211150446961591, −2.11430303594357086416553983875, −2.05971571441448430910075645070, −1.94429569771089455934424791282, −1.90578955481029503645851528710, −1.52519391527340899580865129749, −1.35321873765316198875998271096, −1.30295903260466402709227821052, −1.07898005098471423464073347954, −1.06585308055232875366273062727, −0.909859101426476994500068359206, −0.896810696893167347814059235863, −0.76665215685558266321817816891, 0.76665215685558266321817816891, 0.896810696893167347814059235863, 0.909859101426476994500068359206, 1.06585308055232875366273062727, 1.07898005098471423464073347954, 1.30295903260466402709227821052, 1.35321873765316198875998271096, 1.52519391527340899580865129749, 1.90578955481029503645851528710, 1.94429569771089455934424791282, 2.05971571441448430910075645070, 2.11430303594357086416553983875, 2.32067262049788211150446961591, 2.51016142409128919257421190712, 2.64396345573457361494595660662, 2.87679124674465301627324536735, 2.91933398413382598881907258715, 2.99750412133669500201882345690, 3.11125772861780942345976302348, 3.18872286981222964428811767547, 3.27088980362568678000930031419, 3.64220783691536117804935198785, 3.67912524995280702781805175386, 3.68834038120134269828626261613, 3.74525691457417818661471107468

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

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