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Properties

Label 16-1800e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.102\times 10^{26}$
Sign $1$
Analytic cond. $0.424069$
Root an. cond. $0.947795$
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  − 12·11-s + 16-s + 81-s + 74·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 12·176-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 + ⋯
L(s)  = 1  − 12·11-s + 16-s + 81-s + 74·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 12·176-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 3^{16} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(0.424069\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 3^{16} \cdot 5^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1647576752\)
\(L(\frac12)\) \(\approx\) \(0.1647576752\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{4} + T^{8} \)
3 \( 1 - T^{4} + T^{8} \)
5 \( 1 \)
good7 \( ( 1 - T^{4} + T^{8} )^{2} \)
11 \( ( 1 + T )^{8}( 1 + T + T^{2} )^{4} \)
13 \( ( 1 - T^{4} + T^{8} )^{2} \)
17 \( ( 1 - T^{4} + T^{8} )^{2} \)
19 \( ( 1 + T^{2} )^{8} \)
23 \( ( 1 - T^{4} + T^{8} )^{2} \)
29 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
31 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
37 \( ( 1 + T^{4} )^{4} \)
41 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
43 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
47 \( ( 1 - T^{4} + T^{8} )^{2} \)
53 \( ( 1 + T^{4} )^{4} \)
59 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
61 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
67 \( ( 1 - T^{4} + T^{8} )^{2} \)
71 \( ( 1 + T^{2} )^{8} \)
73 \( ( 1 + T^{4} )^{4} \)
79 \( ( 1 - T^{2} + T^{4} )^{4} \)
83 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
89 \( ( 1 - T^{2} + T^{4} )^{4} \)
97 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
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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

−4.31118157152938583612947943735, −4.07072242480314610154038463632, −3.74276156036104198144548799378, −3.66223817578690572447361199050, −3.60489234103319017187914226199, −3.39229645957378072969942760316, −3.21463747184106147804899495600, −3.15949875164779551818262528937, −3.10752391706199167313223074670, −2.81193297409781177621192577115, −2.77635586702699639491707576588, −2.73882213614474802711523849326, −2.65471491415426207051933902326, −2.62740297154522752529257217227, −2.36556441924893165143430638966, −2.32397788424636289059520841682, −2.18548503439611525922434392025, −2.02345478116842800447309261338, −1.84309416087299561997081507059, −1.73571263724736343362887170463, −1.53070992825207934941872292849, −1.10618018633268612459486766940, −0.61138714427806882444659921441, −0.57918766092620623397402432434, −0.32159476653492231104791788035, 0.32159476653492231104791788035, 0.57918766092620623397402432434, 0.61138714427806882444659921441, 1.10618018633268612459486766940, 1.53070992825207934941872292849, 1.73571263724736343362887170463, 1.84309416087299561997081507059, 2.02345478116842800447309261338, 2.18548503439611525922434392025, 2.32397788424636289059520841682, 2.36556441924893165143430638966, 2.62740297154522752529257217227, 2.65471491415426207051933902326, 2.73882213614474802711523849326, 2.77635586702699639491707576588, 2.81193297409781177621192577115, 3.10752391706199167313223074670, 3.15949875164779551818262528937, 3.21463747184106147804899495600, 3.39229645957378072969942760316, 3.60489234103319017187914226199, 3.66223817578690572447361199050, 3.74276156036104198144548799378, 4.07072242480314610154038463632, 4.31118157152938583612947943735

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

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