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Properties

Label 16-55e16-1.1-c0e8-0-2
Degree $16$
Conductor $7.011\times 10^{27}$
Sign $1$
Analytic cond. $26.9811$
Root an. cond. $1.22868$
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  − 2·3-s + 2·9-s + 16-s + 2·23-s + 25-s − 2·27-s + 2·37-s − 2·47-s − 2·48-s + 2·53-s − 2·67-s − 4·69-s − 4·71-s − 2·75-s + 81-s − 8·97-s + 2·103-s − 4·111-s + 2·113-s + 127-s + 131-s + 137-s + 139-s + 4·141-s + 2·144-s + 149-s + 151-s + ⋯
L(s)  = 1  − 2·3-s + 2·9-s + 16-s + 2·23-s + 25-s − 2·27-s + 2·37-s − 2·47-s − 2·48-s + 2·53-s − 2·67-s − 4·69-s − 4·71-s − 2·75-s + 81-s − 8·97-s + 2·103-s − 4·111-s + 2·113-s + 127-s + 131-s + 137-s + 139-s + 4·141-s + 2·144-s + 149-s + 151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{16} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(26.9811\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{16} \cdot 11^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4478401245\)
\(L(\frac12)\) \(\approx\) \(0.4478401245\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
11 \( 1 \)
good2 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
3 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
7 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
13 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
17 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
41 \( ( 1 + T^{2} )^{8} \)
43 \( ( 1 + T^{4} )^{4} \)
47 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
53 \( ( 1 + T^{2} )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
67 \( ( 1 + T^{2} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
71 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
73 \( ( 1 + T^{4} )^{4} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
83 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
89 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
97 \( ( 1 + T )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
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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

−3.93296716953408260515976340643, −3.86875195280195535636325971538, −3.79321238149767573855030848731, −3.28722024088792694026511042561, −3.28163721768705703274966900633, −3.23202721853379568557912414901, −3.10122238820921117308334786008, −3.01304408927914131749303372989, −2.93712937216136342141289722213, −2.80567334368150643356072340119, −2.77724313699622998023210237044, −2.54927966826502893041674817878, −2.47368658231474662610041889625, −2.16575676738601209133418990327, −2.10502158514400259403998925656, −1.90935833388906729191783013915, −1.85320374243498700534254792946, −1.44253445018441261701359419537, −1.39883454688163023224410367790, −1.35364500982676398930349298139, −1.28399064049069889440361574990, −1.14703196732166602721825382194, −0.76614987799105969735255107414, −0.72697308775844914042274682404, −0.25774165824788508797429561501, 0.25774165824788508797429561501, 0.72697308775844914042274682404, 0.76614987799105969735255107414, 1.14703196732166602721825382194, 1.28399064049069889440361574990, 1.35364500982676398930349298139, 1.39883454688163023224410367790, 1.44253445018441261701359419537, 1.85320374243498700534254792946, 1.90935833388906729191783013915, 2.10502158514400259403998925656, 2.16575676738601209133418990327, 2.47368658231474662610041889625, 2.54927966826502893041674817878, 2.77724313699622998023210237044, 2.80567334368150643356072340119, 2.93712937216136342141289722213, 3.01304408927914131749303372989, 3.10122238820921117308334786008, 3.23202721853379568557912414901, 3.28163721768705703274966900633, 3.28722024088792694026511042561, 3.79321238149767573855030848731, 3.86875195280195535636325971538, 3.93296716953408260515976340643

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

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