Properties

Label 16-300e8-1.1-c1e8-0-7
Degree $16$
Conductor $6.561\times 10^{19}$
Sign $1$
Analytic cond. $1084.39$
Root an. cond. $1.54774$
Motivic weight $1$
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  − 2·2-s + 3·4-s − 8·8-s + 2·9-s + 9·16-s + 16·17-s − 4·18-s − 6·32-s − 32·34-s + 6·36-s − 36·49-s + 32·53-s − 8·61-s + 11·64-s + 48·68-s − 16·72-s + 2·81-s + 72·98-s − 64·106-s + 40·109-s − 112·113-s − 48·121-s + 16·122-s + 127-s + 12·128-s + 131-s − 128·136-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 2.82·8-s + 2/3·9-s + 9/4·16-s + 3.88·17-s − 0.942·18-s − 1.06·32-s − 5.48·34-s + 36-s − 5.14·49-s + 4.39·53-s − 1.02·61-s + 11/8·64-s + 5.82·68-s − 1.88·72-s + 2/9·81-s + 7.27·98-s − 6.21·106-s + 3.83·109-s − 10.5·113-s − 4.36·121-s + 1.44·122-s + 0.0887·127-s + 1.06·128-s + 0.0873·131-s − 10.9·136-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(1084.39\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.227255684\)
\(L(\frac12)\) \(\approx\) \(1.227255684\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + p T^{3} + p^{2} T^{4} )^{2} \)
3 \( 1 - 2 T^{2} + 2 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \)
5 \( 1 \)
good7 \( ( 1 + 18 T^{2} + 162 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 24 T^{2} + 318 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - 16 T^{2} + 334 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( ( 1 - 2 T + p T^{2} )^{8} \)
19 \( ( 1 - 56 T^{2} + 1438 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 - 34 T^{2} + 514 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 80 T^{2} + 3214 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 96 T^{2} + 4158 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 - 112 T^{2} + 5806 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
41 \( ( 1 - 112 T^{2} + 5886 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 + 110 T^{2} + 5890 T^{4} + 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 - 178 T^{2} + 12322 T^{4} - 178 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \)
59 \( ( 1 - 16 T^{2} + 1518 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 2 T + 106 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \)
67 \( ( 1 + 222 T^{2} + 21282 T^{4} + 222 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
71 \( ( 1 + 228 T^{2} + 22806 T^{4} + 228 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{4} \)
79 \( ( 1 - 168 T^{2} + 19470 T^{4} - 168 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 - 82 T^{2} + 4834 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 212 T^{2} + 25990 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{4} \)
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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

−5.26644388986586980497562474462, −5.18914893423673053298608777410, −5.08684597800783301178237091243, −4.92340706773048346020437575434, −4.88818243299042257718981437875, −4.54781125452816927495028166260, −4.14397859670435600432827377860, −4.02174634222205364399853565857, −4.00553694726935926490254029443, −3.96792531876273576665322465797, −3.66108268567277585393370027894, −3.46411383995866646948469270351, −3.25303997592987683850330645766, −3.16828293567293749481697978838, −2.95419845830652817350369686750, −2.76759494464878217451822666490, −2.52278045069656088771437042037, −2.48749072182675905645233202290, −2.42290555944526883482553778518, −1.60225717654946378237550656546, −1.60059385969909256197610543930, −1.44329614820325290940510659740, −1.43874862660615158718974852376, −0.64316686514156208293603579266, −0.52152372064319832357001784615, 0.52152372064319832357001784615, 0.64316686514156208293603579266, 1.43874862660615158718974852376, 1.44329614820325290940510659740, 1.60059385969909256197610543930, 1.60225717654946378237550656546, 2.42290555944526883482553778518, 2.48749072182675905645233202290, 2.52278045069656088771437042037, 2.76759494464878217451822666490, 2.95419845830652817350369686750, 3.16828293567293749481697978838, 3.25303997592987683850330645766, 3.46411383995866646948469270351, 3.66108268567277585393370027894, 3.96792531876273576665322465797, 4.00553694726935926490254029443, 4.02174634222205364399853565857, 4.14397859670435600432827377860, 4.54781125452816927495028166260, 4.88818243299042257718981437875, 4.92340706773048346020437575434, 5.08684597800783301178237091243, 5.18914893423673053298608777410, 5.26644388986586980497562474462

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

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