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Properties

Label 16-1155e8-1.1-c0e8-0-0
Degree $16$
Conductor $3.167\times 10^{24}$
Sign $1$
Analytic cond. $0.0121874$
Root an. cond. $0.759223$
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  − 2·4-s − 4·9-s − 2·11-s + 16-s + 25-s + 4·29-s + 8·36-s + 4·44-s + 49-s + 10·81-s + 8·99-s − 2·100-s − 8·116-s + 121-s + 127-s + 131-s + 137-s + 139-s − 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s − 2·176-s + ⋯
L(s)  = 1  − 2·4-s − 4·9-s − 2·11-s + 16-s + 25-s + 4·29-s + 8·36-s + 4·44-s + 49-s + 10·81-s + 8·99-s − 2·100-s − 8·116-s + 121-s + 127-s + 131-s + 137-s + 139-s − 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s − 2·176-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 5^{8} \cdot 7^{8} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(0.0121874\)
Root analytic conductor: \(0.759223\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 5^{8} \cdot 7^{8} \cdot 11^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1390490563\)
\(L(\frac12)\) \(\approx\) \(0.1390490563\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T^{2} )^{4} \)
5 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
7 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
11 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
good2 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
13 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
17 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
19 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
23 \( ( 1 + T^{2} )^{8} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
43 \( ( 1 + T^{2} )^{8} \)
47 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
53 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
67 \( ( 1 - T )^{8}( 1 + T )^{8} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
73 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
83 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
89 \( ( 1 + T^{2} )^{8} \)
97 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
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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

−4.40779551606272287050418061375, −4.34170440711450924310729035425, −4.26410260718424871163215445963, −4.19903367632250260719653735966, −3.81278511791054016301068706905, −3.69891385183011610472171986263, −3.60757696700878178513388455860, −3.43936272438415177248638078228, −3.43192396783668112536100469165, −3.33367096344250520956081290293, −3.03675516905852479475590265397, −2.87864373818851660644446417177, −2.79679686579484575289669116450, −2.68254238231659724828567889576, −2.54687174461918650407629453599, −2.53631638899935148784645484442, −2.31126491365550684933553375934, −2.23069044410926968136325685752, −2.20863429100374431803230802104, −1.57756135651650738842978431603, −1.51222420174332347612571282051, −1.17991437844168444457522533349, −1.03573948806927895095104719075, −0.64427114098009765078116628240, −0.38632136365402133628361192920, 0.38632136365402133628361192920, 0.64427114098009765078116628240, 1.03573948806927895095104719075, 1.17991437844168444457522533349, 1.51222420174332347612571282051, 1.57756135651650738842978431603, 2.20863429100374431803230802104, 2.23069044410926968136325685752, 2.31126491365550684933553375934, 2.53631638899935148784645484442, 2.54687174461918650407629453599, 2.68254238231659724828567889576, 2.79679686579484575289669116450, 2.87864373818851660644446417177, 3.03675516905852479475590265397, 3.33367096344250520956081290293, 3.43192396783668112536100469165, 3.43936272438415177248638078228, 3.60757696700878178513388455860, 3.69891385183011610472171986263, 3.81278511791054016301068706905, 4.19903367632250260719653735966, 4.26410260718424871163215445963, 4.34170440711450924310729035425, 4.40779551606272287050418061375

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

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