Properties

Label 16-605e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.795\times 10^{22}$
Sign $1$
Analytic cond. $6.90716\times 10^{-5}$
Root an. cond. $0.549485$
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 − 8·23-s + 25-s − 2·27-s + 2·37-s − 2·47-s − 2·48-s + 2·53-s + 8·67-s + 16·69-s − 4·71-s − 2·75-s + 81-s + 2·97-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 − 8·23-s + 25-s − 2·27-s + 2·37-s − 2·47-s − 2·48-s + 2·53-s + 8·67-s + 16·69-s − 4·71-s − 2·75-s + 81-s + 2·97-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

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

Invariants

Degree: \(16\)
Conductor: \(5^{8} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(6.90716\times 10^{-5}\)
Root analytic conductor: \(0.549485\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{8} \cdot 11^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1444532389\)
\(L(\frac12)\) \(\approx\) \(0.1444532389\)
\(L(1)\) 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 \( ( 1 + T )^{8}( 1 + T^{2} )^{4} \)
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 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
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 \( ( 1 - T )^{8}( 1 + T^{2} )^{4} \)
71 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
73 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
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 \( ( 1 + T^{2} )^{8} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
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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

−4.95825131402453036523765982036, −4.70570813453337572570895833179, −4.60615088214034175762575072142, −4.48087983787020986270055012372, −4.45377581246930556150005202452, −4.27567571512133134146275240000, −3.95671247726461791859387824377, −3.94135349979347231119537022174, −3.89255125114819955002376219707, −3.87965630933932625036981791227, −3.64250115166442234441480454658, −3.35058899478422628591489559666, −3.30617896332230015490208542244, −3.05004765402552925916788092080, −3.04003834448585699471247501327, −2.51612685095028633135248256053, −2.43553094935647101183460505084, −2.24008365969667372946248075787, −2.04423283722651196890436394849, −1.97738804122392864361554238037, −1.93346830409320891915823061122, −1.74476970960032848409458642893, −1.14395072034265558483809096537, −1.08259507175776341662098159118, −0.63173197566369554946515770389, 0.63173197566369554946515770389, 1.08259507175776341662098159118, 1.14395072034265558483809096537, 1.74476970960032848409458642893, 1.93346830409320891915823061122, 1.97738804122392864361554238037, 2.04423283722651196890436394849, 2.24008365969667372946248075787, 2.43553094935647101183460505084, 2.51612685095028633135248256053, 3.04003834448585699471247501327, 3.05004765402552925916788092080, 3.30617896332230015490208542244, 3.35058899478422628591489559666, 3.64250115166442234441480454658, 3.87965630933932625036981791227, 3.89255125114819955002376219707, 3.94135349979347231119537022174, 3.95671247726461791859387824377, 4.27567571512133134146275240000, 4.45377581246930556150005202452, 4.48087983787020986270055012372, 4.60615088214034175762575072142, 4.70570813453337572570895833179, 4.95825131402453036523765982036

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

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