Properties

Label 16-1620e8-1.1-c0e8-0-2
Degree $16$
Conductor $4.744\times 10^{25}$
Sign $1$
Analytic cond. $0.182548$
Root an. cond. $0.899158$
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  + 4·13-s + 16-s − 8·37-s + 8·73-s − 4·97-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 4·208-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 4·13-s + 16-s − 8·37-s + 8·73-s − 4·97-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 4·208-s + 211-s + 223-s + 227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{32} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(0.182548\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1620} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{32} \cdot 5^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.507270723\)
\(L(\frac12)\) \(\approx\) \(1.507270723\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{4} + T^{8} \)
3 \( 1 \)
5 \( 1 - T^{4} + T^{8} \)
good7 \( ( 1 - T^{4} + T^{8} )^{2} \)
11 \( ( 1 - T^{2} + T^{4} )^{4} \)
13 \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
17 \( ( 1 + T^{4} )^{4} \)
19 \( ( 1 + T^{2} )^{8} \)
23 \( ( 1 - T^{4} + T^{8} )^{2} \)
29 \( ( 1 - T^{4} + T^{8} )^{2} \)
31 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
37 \( ( 1 + T )^{8}( 1 + T^{2} )^{4} \)
41 \( ( 1 - T^{4} + T^{8} )^{2} \)
43 \( ( 1 - T^{4} + T^{8} )^{2} \)
47 \( ( 1 - T^{4} + T^{8} )^{2} \)
53 \( ( 1 + T^{4} )^{4} \)
59 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
61 \( ( 1 - T^{2} + T^{4} )^{4} \)
67 \( ( 1 - T^{4} + T^{8} )^{2} \)
71 \( ( 1 + T^{2} )^{8} \)
73 \( ( 1 - T )^{8}( 1 + T^{2} )^{4} \)
79 \( ( 1 - T^{2} + T^{4} )^{4} \)
83 \( ( 1 - T^{4} + T^{8} )^{2} \)
89 \( ( 1 + T^{4} )^{4} \)
97 \( ( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
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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

−4.02741965393529735616483427007, −4.00772329631619556746858186390, −3.99234044952871783814205997437, −3.81187737912489637959534621995, −3.67997943357391030786871549092, −3.59355093683939764285563496747, −3.56058403767100493915468677189, −3.30024750514592060519041955445, −3.15068881451251900446742087617, −3.14060269571621754647289784448, −3.11911721403236545769765482694, −3.05051686259952760547042442518, −2.71143388325920194190630572401, −2.38256177643502493159938384063, −2.26679353649030354655236645600, −2.19970194892259404969061869648, −1.98351477667380512182141178703, −1.92965298235769861204746495724, −1.70232108226614262334799508408, −1.61856792214718953797928934948, −1.43293737938939737523629963426, −1.20796540875959255527920609997, −1.13680127332811590148304639570, −0.893379945434769493511116456942, −0.56277024507619513247984376814, 0.56277024507619513247984376814, 0.893379945434769493511116456942, 1.13680127332811590148304639570, 1.20796540875959255527920609997, 1.43293737938939737523629963426, 1.61856792214718953797928934948, 1.70232108226614262334799508408, 1.92965298235769861204746495724, 1.98351477667380512182141178703, 2.19970194892259404969061869648, 2.26679353649030354655236645600, 2.38256177643502493159938384063, 2.71143388325920194190630572401, 3.05051686259952760547042442518, 3.11911721403236545769765482694, 3.14060269571621754647289784448, 3.15068881451251900446742087617, 3.30024750514592060519041955445, 3.56058403767100493915468677189, 3.59355093683939764285563496747, 3.67997943357391030786871549092, 3.81187737912489637959534621995, 3.99234044952871783814205997437, 4.00772329631619556746858186390, 4.02741965393529735616483427007

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

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