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Properties

Label 16-2420e8-1.1-c0e8-0-2
Degree $16$
Conductor $1.176\times 10^{27}$
Sign $1$
Analytic cond. $4.52668$
Root an. cond. $1.09897$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 9-s + 25-s − 2·31-s − 45-s + 2·49-s + 2·59-s + 2·71-s + 81-s + 8·89-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·155-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 5-s + 9-s + 25-s − 2·31-s − 45-s + 2·49-s + 2·59-s + 2·71-s + 81-s + 8·89-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·155-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 5^{8} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(4.52668\)
Root analytic conductor: \(1.09897\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 5^{8} \cdot 11^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.929619266\)
\(L(\frac12)\) \(\approx\) \(1.929619266\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \)
11 \( 1 \)
good3 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
7 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
13 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
17 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
23 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
37 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
59 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
67 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
71 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
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 \( ( 1 - T + T^{2} )^{8} \)
97 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
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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

−3.89521802301024265243597561064, −3.86894818544094382151212380046, −3.81974178859127751986901353681, −3.56140749984616635199153066956, −3.50283287176845941773047454822, −3.48949480801157832884731137085, −3.28623945245019740050422165235, −3.23743971907222525975417923784, −2.90887961881098841154525553775, −2.88364228435537577449024523438, −2.71272961094953116670570909071, −2.65805700122582981417675332536, −2.65301150462523256419588941258, −2.11118249477062621422538490668, −2.08524313698285205606652995929, −2.00261291112293776846983326010, −1.98959284496247387596298380905, −1.93065118814933726522908269176, −1.81259670236097633496321265357, −1.37611204560724459688490586341, −1.35555873613772083650341355257, −0.841891944137098416841549083779, −0.823642899700743199778925897213, −0.807548678767695024771952773920, −0.62817984632745541366758465576, 0.62817984632745541366758465576, 0.807548678767695024771952773920, 0.823642899700743199778925897213, 0.841891944137098416841549083779, 1.35555873613772083650341355257, 1.37611204560724459688490586341, 1.81259670236097633496321265357, 1.93065118814933726522908269176, 1.98959284496247387596298380905, 2.00261291112293776846983326010, 2.08524313698285205606652995929, 2.11118249477062621422538490668, 2.65301150462523256419588941258, 2.65805700122582981417675332536, 2.71272961094953116670570909071, 2.88364228435537577449024523438, 2.90887961881098841154525553775, 3.23743971907222525975417923784, 3.28623945245019740050422165235, 3.48949480801157832884731137085, 3.50283287176845941773047454822, 3.56140749984616635199153066956, 3.81974178859127751986901353681, 3.86894818544094382151212380046, 3.89521802301024265243597561064

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

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