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Properties

Label 16-3360e8-1.1-c0e8-0-4
Degree $16$
Conductor $1.624\times 10^{28}$
Sign $1$
Analytic cond. $62.5131$
Root an. cond. $1.29493$
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  − 8·23-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + 257-s + ⋯
L(s)  = 1  − 8·23-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + 257-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(62.5131\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.047755527\)
\(L(\frac12)\) \(\approx\) \(1.047755527\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T^{8} \)
5 \( 1 + T^{8} \)
7 \( ( 1 + T^{4} )^{2} \)
good11 \( ( 1 + T^{2} )^{8} \)
13 \( ( 1 + T^{8} )^{2} \)
17 \( ( 1 + T^{4} )^{4} \)
19 \( ( 1 + T^{8} )^{2} \)
23 \( ( 1 + T )^{8}( 1 + T^{2} )^{4} \)
29 \( ( 1 - T )^{8}( 1 + T )^{8} \)
31 \( ( 1 - T )^{8}( 1 + T )^{8} \)
37 \( ( 1 + T^{4} )^{4} \)
41 \( ( 1 + T^{2} )^{8} \)
43 \( ( 1 + T^{4} )^{4} \)
47 \( ( 1 + T^{4} )^{4} \)
53 \( ( 1 + T^{4} )^{4} \)
59 \( ( 1 + T^{8} )^{2} \)
61 \( ( 1 + T^{8} )^{2} \)
67 \( ( 1 + T^{4} )^{4} \)
71 \( ( 1 + T^{4} )^{4} \)
73 \( ( 1 + T^{4} )^{4} \)
79 \( ( 1 + T^{4} )^{4} \)
83 \( ( 1 + T^{8} )^{2} \)
89 \( ( 1 - T )^{8}( 1 + T )^{8} \)
97 \( ( 1 + T^{4} )^{4} \)
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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

−3.80189580948706992139046807584, −3.69973904004294307132832688393, −3.69627059804203428995252987243, −3.48060568041013044083468479434, −3.34844838304635508756046542305, −3.15820378983258923510131805670, −3.13208003276914848998773191894, −2.98165141913523582906484344811, −2.76303946059306675947502956107, −2.72254477392863742905442347963, −2.64455850369572046137065960068, −2.41571812206722988649321887569, −2.28981581832743426789709247341, −2.07791087825325673859182953490, −2.02104356333769841329614070332, −1.89598614928410381854738811445, −1.89008385081340470696412097661, −1.88016560052046086808786165243, −1.60027795934095171860718553013, −1.45164395205423187010011638461, −1.32075008455084012930693112558, −1.07697561750994030661509749579, −0.66058366343829626933164759105, −0.44975563607529380306180208946, −0.44966988908178089992122292879, 0.44966988908178089992122292879, 0.44975563607529380306180208946, 0.66058366343829626933164759105, 1.07697561750994030661509749579, 1.32075008455084012930693112558, 1.45164395205423187010011638461, 1.60027795934095171860718553013, 1.88016560052046086808786165243, 1.89008385081340470696412097661, 1.89598614928410381854738811445, 2.02104356333769841329614070332, 2.07791087825325673859182953490, 2.28981581832743426789709247341, 2.41571812206722988649321887569, 2.64455850369572046137065960068, 2.72254477392863742905442347963, 2.76303946059306675947502956107, 2.98165141913523582906484344811, 3.13208003276914848998773191894, 3.15820378983258923510131805670, 3.34844838304635508756046542305, 3.48060568041013044083468479434, 3.69627059804203428995252987243, 3.69973904004294307132832688393, 3.80189580948706992139046807584

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

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