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Properties

Label 16-3000e8-1.1-c0e8-0-5
Degree $16$
Conductor $6.561\times 10^{27}$
Sign $1$
Analytic cond. $25.2480$
Root an. cond. $1.22359$
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  + 4-s + 9-s + 6·11-s + 4·29-s + 6·31-s + 36-s + 6·44-s + 2·49-s − 6·59-s + 4·79-s + 6·99-s − 4·101-s + 4·116-s + 21·121-s + 6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 4-s + 9-s + 6·11-s + 4·29-s + 6·31-s + 36-s + 6·44-s + 2·49-s − 6·59-s + 4·79-s + 6·99-s − 4·101-s + 4·116-s + 21·121-s + 6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 3^{8} \cdot 5^{24}\)
Sign: $1$
Analytic conductor: \(25.2480\)
Root analytic conductor: \(1.22359\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 3^{8} \cdot 5^{24} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(10.36897594\)
\(L(\frac12)\) \(\approx\) \(10.36897594\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
3 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
5 \( 1 \)
good7 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
11 \( ( 1 - T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
31 \( ( 1 - T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
37 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
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 \( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
67 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
83 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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

−3.98131245235645868755830743702, −3.64192916736800699075887973681, −3.63928840800091695825325124516, −3.51913509247070178003888219752, −3.36175417317806411162937860425, −3.15252157024062543783996654486, −3.14006485253258046542568761449, −2.94744380740675787822612735859, −2.93866743468009594989839000296, −2.90957621706881579655095283458, −2.76584652263129322531534567422, −2.38753895428805540666832907635, −2.34574353190113118426886878855, −2.25853929159703558101205277614, −2.18739472201456455980103222435, −1.94894984183502276494461934129, −1.89268259881723930247573916671, −1.55883008556248954393484735442, −1.37787492530949120533056042002, −1.37059082607324065602644766319, −1.10091802033193046010119143382, −1.07613183304181199692817029994, −1.04748391440211453046418743081, −1.00312013691430597831263110279, −0.813423702081382022553292115054, 0.813423702081382022553292115054, 1.00312013691430597831263110279, 1.04748391440211453046418743081, 1.07613183304181199692817029994, 1.10091802033193046010119143382, 1.37059082607324065602644766319, 1.37787492530949120533056042002, 1.55883008556248954393484735442, 1.89268259881723930247573916671, 1.94894984183502276494461934129, 2.18739472201456455980103222435, 2.25853929159703558101205277614, 2.34574353190113118426886878855, 2.38753895428805540666832907635, 2.76584652263129322531534567422, 2.90957621706881579655095283458, 2.93866743468009594989839000296, 2.94744380740675787822612735859, 3.14006485253258046542568761449, 3.15252157024062543783996654486, 3.36175417317806411162937860425, 3.51913509247070178003888219752, 3.63928840800091695825325124516, 3.64192916736800699075887973681, 3.98131245235645868755830743702

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

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