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Properties

Label 16-75e16-1.1-c1e8-0-10
Degree $16$
Conductor $1.002\times 10^{30}$
Sign $1$
Analytic cond. $1.65652\times 10^{13}$
Root an. cond. $6.70192$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $8$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·4-s − 10·13-s + 14·16-s − 18·19-s − 26·31-s + 40·43-s − 36·49-s + 60·52-s − 24·61-s − 15·64-s + 40·67-s − 40·73-s + 108·76-s − 42·79-s + 40·97-s − 10·103-s − 22·109-s − 53·121-s + 156·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 3·4-s − 2.77·13-s + 7/2·16-s − 4.12·19-s − 4.66·31-s + 6.09·43-s − 5.14·49-s + 8.32·52-s − 3.07·61-s − 1.87·64-s + 4.88·67-s − 4.68·73-s + 12.3·76-s − 4.72·79-s + 4.06·97-s − 0.985·103-s − 2.10·109-s − 4.81·121-s + 14.0·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 5^{32}\)
Sign: $1$
Analytic conductor: \(1.65652\times 10^{13}\)
Root analytic conductor: \(6.70192\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(8\)
Selberg data: \((16,\ 3^{16} \cdot 5^{32} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + 3 p T^{2} + 11 p T^{4} + 63 T^{6} + 145 T^{8} + 63 p^{2} T^{10} + 11 p^{5} T^{12} + 3 p^{7} T^{14} + p^{8} T^{16} \)
7 \( ( 1 + 18 T^{2} - 5 T^{3} + 159 T^{4} - 5 p T^{5} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( 1 + 53 T^{2} + 1478 T^{4} + 27111 T^{6} + 350695 T^{8} + 27111 p^{2} T^{10} + 1478 p^{4} T^{12} + 53 p^{6} T^{14} + p^{8} T^{16} \)
13 \( ( 1 + 5 T + 42 T^{2} + 135 T^{3} + 709 T^{4} + 135 p T^{5} + 42 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( 1 + 86 T^{2} + 3782 T^{4} + 108063 T^{6} + 2170345 T^{8} + 108063 p^{2} T^{10} + 3782 p^{4} T^{12} + 86 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 + 9 T + 87 T^{2} + 452 T^{3} + 2475 T^{4} + 452 p T^{5} + 87 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 + 89 T^{2} + 4652 T^{4} + 163437 T^{6} + 4345135 T^{8} + 163437 p^{2} T^{10} + 4652 p^{4} T^{12} + 89 p^{6} T^{14} + p^{8} T^{16} \)
29 \( 1 + 72 T^{2} + 3208 T^{4} + 114009 T^{6} + 3314245 T^{8} + 114009 p^{2} T^{10} + 3208 p^{4} T^{12} + 72 p^{6} T^{14} + p^{8} T^{16} \)
31 \( ( 1 + 13 T + 108 T^{2} + 831 T^{3} + 5575 T^{4} + 831 p T^{5} + 108 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 58 T^{2} + 195 T^{3} + 1459 T^{4} + 195 p T^{5} + 58 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
41 \( 1 + 8 T^{2} + 2198 T^{4} + 124176 T^{6} + 2249095 T^{8} + 124176 p^{2} T^{10} + 2198 p^{4} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} \)
43 \( ( 1 - 20 T + 287 T^{2} - 2730 T^{3} + 20939 T^{4} - 2730 p T^{5} + 287 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 + 276 T^{2} + 35272 T^{4} + 2796573 T^{6} + 154609105 T^{8} + 2796573 p^{2} T^{10} + 35272 p^{4} T^{12} + 276 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 + 114 T^{2} + 8722 T^{4} + 400287 T^{6} + 21803425 T^{8} + 400287 p^{2} T^{10} + 8722 p^{4} T^{12} + 114 p^{6} T^{14} + p^{8} T^{16} \)
59 \( 1 + 27 T^{2} + 7413 T^{4} + 289274 T^{6} + 32570895 T^{8} + 289274 p^{2} T^{10} + 7413 p^{4} T^{12} + 27 p^{6} T^{14} + p^{8} T^{16} \)
61 \( ( 1 + 12 T + 258 T^{2} + 2024 T^{3} + 23715 T^{4} + 2024 p T^{5} + 258 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( ( 1 - 20 T + 273 T^{2} - 2440 T^{3} + 21209 T^{4} - 2440 p T^{5} + 273 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( 1 + 18 T^{2} + 15898 T^{4} + 337341 T^{6} + 109240345 T^{8} + 337341 p^{2} T^{10} + 15898 p^{4} T^{12} + 18 p^{6} T^{14} + p^{8} T^{16} \)
73 \( ( 1 + 20 T + 207 T^{2} + 1290 T^{3} + 8719 T^{4} + 1290 p T^{5} + 207 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + 21 T + 352 T^{2} + 3783 T^{3} + 37825 T^{4} + 3783 p T^{5} + 352 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 + 504 T^{2} + 117742 T^{4} + 16975632 T^{6} + 1677152095 T^{8} + 16975632 p^{2} T^{10} + 117742 p^{4} T^{12} + 504 p^{6} T^{14} + p^{8} T^{16} \)
89 \( 1 + 287 T^{2} + 31838 T^{4} + 1246389 T^{6} - 2224505 T^{8} + 1246389 p^{2} T^{10} + 31838 p^{4} T^{12} + 287 p^{6} T^{14} + p^{8} T^{16} \)
97 \( ( 1 - 20 T + 338 T^{2} - 3480 T^{3} + 37459 T^{4} - 3480 p T^{5} + 338 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−3.79216914426158551440719689073, −3.56900759902360852796645248102, −3.49934306125849376318714492324, −3.47914735020155176200824879727, −3.45157047508483709328766657855, −3.43780516765246022412443126066, −3.19307716242613237605822459817, −2.98613536615495087464121461111, −2.70756790372355922518065044348, −2.66477694049925770816041155032, −2.53688349692056391704307645428, −2.52174688565089569720757644389, −2.39106372204962878119590993892, −2.34200164420243306976839916093, −2.19566466421410058860399429236, −2.19334972175916529556959223547, −2.12880017383867173891476776818, −1.79145443924952130023615130082, −1.66992542119339020024071910740, −1.48119130725089554983401830008, −1.25334630969181260829377186540, −1.23037346324034270374394777470, −1.07299203935134275474523373725, −1.06552298075319220856487162639, −1.02940057751921054231000198059, 0, 0, 0, 0, 0, 0, 0, 0, 1.02940057751921054231000198059, 1.06552298075319220856487162639, 1.07299203935134275474523373725, 1.23037346324034270374394777470, 1.25334630969181260829377186540, 1.48119130725089554983401830008, 1.66992542119339020024071910740, 1.79145443924952130023615130082, 2.12880017383867173891476776818, 2.19334972175916529556959223547, 2.19566466421410058860399429236, 2.34200164420243306976839916093, 2.39106372204962878119590993892, 2.52174688565089569720757644389, 2.53688349692056391704307645428, 2.66477694049925770816041155032, 2.70756790372355922518065044348, 2.98613536615495087464121461111, 3.19307716242613237605822459817, 3.43780516765246022412443126066, 3.45157047508483709328766657855, 3.47914735020155176200824879727, 3.49934306125849376318714492324, 3.56900759902360852796645248102, 3.79216914426158551440719689073

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

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