Properties

Label 16-1400e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.476\times 10^{25}$
Sign $1$
Analytic cond. $0.0567912$
Root an. cond. $0.835877$
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  + 2·2-s + 4-s − 8·7-s − 3·9-s − 16·14-s − 6·18-s − 4·23-s + 25-s − 8·28-s − 2·32-s − 3·36-s − 8·46-s + 36·49-s + 2·50-s + 24·63-s − 4·64-s + 4·71-s − 6·79-s + 6·81-s − 4·92-s + 72·98-s + 100-s − 6·113-s − 2·121-s + 48·126-s + 127-s − 2·128-s + ⋯
L(s)  = 1  + 2·2-s + 4-s − 8·7-s − 3·9-s − 16·14-s − 6·18-s − 4·23-s + 25-s − 8·28-s − 2·32-s − 3·36-s − 8·46-s + 36·49-s + 2·50-s + 24·63-s − 4·64-s + 4·71-s − 6·79-s + 6·81-s − 4·92-s + 72·98-s + 100-s − 6·113-s − 2·121-s + 48·126-s + 127-s − 2·128-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 5^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(0.0567912\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02886826731\)
\(L(\frac12)\) \(\approx\) \(0.02886826731\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
5 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
7 \( ( 1 + T )^{8} \)
good3 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
13 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
19 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
23 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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 \( ( 1 - T )^{8}( 1 + T )^{8} \)
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 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
61 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
79 \( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
83 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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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

−4.19190532648003138063654156884, −4.04464906417913555312352078416, −4.00035878326675919081495598049, −3.99046165098812142903869231700, −3.82758548451803870063428620526, −3.59567777942205648497474576155, −3.49943729842409964607718231947, −3.48607617149681610483187808566, −3.25838588822926833516245191684, −3.25653215320504421441930454776, −3.12310501507340961386727973010, −2.87030431347553933941278850483, −2.84926056531156137955195571858, −2.80773861137065744452294652644, −2.58009072561909172004609424629, −2.43838579966832259913146502127, −2.35155292541466349688018106358, −2.22733079856664695251102372414, −2.22580581041697133799386198089, −1.86298532243176376511015882646, −1.38374054779614435746599674834, −1.12233116392352343792814609319, −1.04958346167430216007517007516, −0.26003480689122438151243226334, −0.21081003325642543076509382466, 0.21081003325642543076509382466, 0.26003480689122438151243226334, 1.04958346167430216007517007516, 1.12233116392352343792814609319, 1.38374054779614435746599674834, 1.86298532243176376511015882646, 2.22580581041697133799386198089, 2.22733079856664695251102372414, 2.35155292541466349688018106358, 2.43838579966832259913146502127, 2.58009072561909172004609424629, 2.80773861137065744452294652644, 2.84926056531156137955195571858, 2.87030431347553933941278850483, 3.12310501507340961386727973010, 3.25653215320504421441930454776, 3.25838588822926833516245191684, 3.48607617149681610483187808566, 3.49943729842409964607718231947, 3.59567777942205648497474576155, 3.82758548451803870063428620526, 3.99046165098812142903869231700, 4.00035878326675919081495598049, 4.04464906417913555312352078416, 4.19190532648003138063654156884

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

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