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Properties

Label 16-10e32-1.1-c1e8-0-9
Degree $16$
Conductor $1.000\times 10^{32}$
Sign $1$
Analytic cond. $1.65278\times 10^{15}$
Root an. cond. $8.93590$
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  − 13·9-s + 10·11-s + 12·19-s − 32·29-s + 2·31-s − 42·41-s − 35·49-s + 24·59-s − 34·61-s − 4·71-s − 4·79-s + 79·81-s − 58·89-s − 130·99-s − 72·101-s − 34·109-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 58·169-s + ⋯
L(s)  = 1  − 4.33·9-s + 3.01·11-s + 2.75·19-s − 5.94·29-s + 0.359·31-s − 6.55·41-s − 5·49-s + 3.12·59-s − 4.35·61-s − 0.474·71-s − 0.450·79-s + 79/9·81-s − 6.14·89-s − 13.0·99-s − 7.16·101-s − 3.25·109-s − 0.272·121-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 + 0.0773·167-s − 4.46·169-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 5^{32}\)
Sign: $1$
Analytic conductor: \(1.65278\times 10^{15}\)
Root analytic conductor: \(8.93590\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(8\)
Selberg data: \((16,\ 2^{32} \cdot 5^{32} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

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)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + 13 T^{2} + 10 p^{2} T^{4} + 428 T^{6} + 1489 T^{8} + 428 p^{2} T^{10} + 10 p^{6} T^{12} + 13 p^{6} T^{14} + p^{8} T^{16} \)
7 \( 1 + 5 p T^{2} + 551 T^{4} + 775 p T^{6} + 41176 T^{8} + 775 p^{3} T^{10} + 551 p^{4} T^{12} + 5 p^{7} T^{14} + p^{8} T^{16} \)
11 \( ( 1 - 5 T + 39 T^{2} - 125 T^{3} + 596 T^{4} - 125 p T^{5} + 39 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 + 58 T^{2} + 1775 T^{4} + 36828 T^{6} + 555429 T^{8} + 36828 p^{2} T^{10} + 1775 p^{4} T^{12} + 58 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 + 47 T^{2} + 1855 T^{4} + 43857 T^{6} + 903824 T^{8} + 43857 p^{2} T^{10} + 1855 p^{4} T^{12} + 47 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 - 6 T + 37 T^{2} - 118 T^{3} + 695 T^{4} - 118 p T^{5} + 37 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 + 65 T^{2} + 2906 T^{4} + 97420 T^{6} + 2436721 T^{8} + 97420 p^{2} T^{10} + 2906 p^{4} T^{12} + 65 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 + 16 T + 177 T^{2} + 1368 T^{3} + 8585 T^{4} + 1368 p T^{5} + 177 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - T + 40 T^{2} + 246 T^{3} + 479 T^{4} + 246 p T^{5} + 40 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 + 145 T^{2} + 11926 T^{4} + 685260 T^{6} + 29054021 T^{8} + 685260 p^{2} T^{10} + 11926 p^{4} T^{12} + 145 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 + 21 T + 315 T^{2} + 2999 T^{3} + 22784 T^{4} + 2999 p T^{5} + 315 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 + 219 T^{2} + 21607 T^{4} + 1334737 T^{6} + 62871560 T^{8} + 1334737 p^{2} T^{10} + 21607 p^{4} T^{12} + 219 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 + 142 T^{2} + 10995 T^{4} + 696812 T^{6} + 36724229 T^{8} + 696812 p^{2} T^{10} + 10995 p^{4} T^{12} + 142 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 + 225 T^{2} + 20366 T^{4} + 968700 T^{6} + 39595981 T^{8} + 968700 p^{2} T^{10} + 20366 p^{4} T^{12} + 225 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 - 12 T + 255 T^{2} - 1962 T^{3} + 22719 T^{4} - 1962 p T^{5} + 255 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 17 T + 338 T^{2} + 54 p T^{3} + 555 p T^{4} + 54 p^{2} T^{5} + 338 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 + 255 T^{2} + 33071 T^{4} + 2932765 T^{6} + 209204376 T^{8} + 2932765 p^{2} T^{10} + 33071 p^{4} T^{12} + 255 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 + 2 T + 93 T^{2} + 234 T^{3} + 3095 T^{4} + 234 p T^{5} + 93 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 + p T^{2} + 9510 T^{4} + 1012388 T^{6} + 67144469 T^{8} + 1012388 p^{2} T^{10} + 9510 p^{4} T^{12} + p^{7} T^{14} + p^{8} T^{16} \)
79 \( ( 1 + 2 T + 125 T^{2} - 1118 T^{3} + 3739 T^{4} - 1118 p T^{5} + 125 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 + 405 T^{2} + 83666 T^{4} + 11300060 T^{6} + 1093918281 T^{8} + 11300060 p^{2} T^{10} + 83666 p^{4} T^{12} + 405 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 29 T + 497 T^{2} + 5777 T^{3} + 58300 T^{4} + 5777 p T^{5} + 497 p^{2} T^{6} + 29 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 + 270 T^{2} + 23651 T^{4} - 505940 T^{6} - 217877739 T^{8} - 505940 p^{2} T^{10} + 23651 p^{4} T^{12} + 270 p^{6} T^{14} + p^{8} T^{16} \)
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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

−3.56425715482825498686794455828, −3.26320134156497164565689249581, −3.14056468897243667379638786992, −3.10045556812992360020565741165, −3.09825439852792099311388497586, −3.00163105764399545233730896387, −2.96694628784115506187257031181, −2.93150948383364771955331377912, −2.93124544345099681874774497643, −2.54593636451941447336317492411, −2.39263449339750313768577715800, −2.29900573311945723566364674924, −2.21922571827260423106447080361, −2.11236949948932865172450890325, −1.96289871721748178048960661225, −1.94219517467993363910939415789, −1.68789492842039055842843871870, −1.66835505820144829877686700105, −1.43705477113310257073422948073, −1.34602493238129097718645756514, −1.31311515854112078199194280381, −1.15468722416862480228470721792, −1.11443349851417808931866131651, −1.08467106330867127577927602856, −1.03427147584083182743603998158, 0, 0, 0, 0, 0, 0, 0, 0, 1.03427147584083182743603998158, 1.08467106330867127577927602856, 1.11443349851417808931866131651, 1.15468722416862480228470721792, 1.31311515854112078199194280381, 1.34602493238129097718645756514, 1.43705477113310257073422948073, 1.66835505820144829877686700105, 1.68789492842039055842843871870, 1.94219517467993363910939415789, 1.96289871721748178048960661225, 2.11236949948932865172450890325, 2.21922571827260423106447080361, 2.29900573311945723566364674924, 2.39263449339750313768577715800, 2.54593636451941447336317492411, 2.93124544345099681874774497643, 2.93150948383364771955331377912, 2.96694628784115506187257031181, 3.00163105764399545233730896387, 3.09825439852792099311388497586, 3.10045556812992360020565741165, 3.14056468897243667379638786992, 3.26320134156497164565689249581, 3.56425715482825498686794455828

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

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