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Properties

Label 16-2300e8-1.1-c1e8-0-1
Degree $16$
Conductor $7.831\times 10^{26}$
Sign $1$
Analytic cond. $1.29431\times 10^{10}$
Root an. cond. $4.28550$
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  + 10·9-s + 2·11-s − 14·19-s + 10·29-s + 28·31-s + 6·41-s + 27·49-s − 22·59-s + 44·61-s + 36·71-s − 50·79-s + 47·81-s + 20·99-s + 38·101-s − 20·109-s − 33·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 41·169-s − 140·171-s + ⋯
L(s)  = 1  + 10/3·9-s + 0.603·11-s − 3.21·19-s + 1.85·29-s + 5.02·31-s + 0.937·41-s + 27/7·49-s − 2.86·59-s + 5.63·61-s + 4.27·71-s − 5.62·79-s + 47/9·81-s + 2.01·99-s + 3.78·101-s − 1.91·109-s − 3·121-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 + 3.15·169-s − 10.7·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 5^{16} \cdot 23^{8}\)
Sign: $1$
Analytic conductor: \(1.29431\times 10^{10}\)
Root analytic conductor: \(4.28550\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 5^{16} \cdot 23^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(9.133371320\)
\(L(\frac12)\) \(\approx\) \(9.133371320\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23 \( ( 1 + T^{2} )^{4} \)
good3 \( 1 - 10 T^{2} + 53 T^{4} - 229 T^{6} + 802 T^{8} - 229 p^{2} T^{10} + 53 p^{4} T^{12} - 10 p^{6} T^{14} + p^{8} T^{16} \)
7 \( 1 - 27 T^{2} + 366 T^{4} - 515 p T^{6} + 28386 T^{8} - 515 p^{3} T^{10} + 366 p^{4} T^{12} - 27 p^{6} T^{14} + p^{8} T^{16} \)
11 \( ( 1 - T + 18 T^{2} - 9 T^{3} + 274 T^{4} - 9 p T^{5} + 18 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 - 41 T^{2} + 641 T^{4} - 4473 T^{6} + 22961 T^{8} - 4473 p^{2} T^{10} + 641 p^{4} T^{12} - 41 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 - 64 T^{2} + 1820 T^{4} - 31104 T^{6} + 469702 T^{8} - 31104 p^{2} T^{10} + 1820 p^{4} T^{12} - 64 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 + 7 T + 68 T^{2} + 339 T^{3} + 1934 T^{4} + 339 p T^{5} + 68 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 5 T + 85 T^{2} - 303 T^{3} + 3155 T^{4} - 303 p T^{5} + 85 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 14 T + 173 T^{2} - 1329 T^{3} + 8794 T^{4} - 1329 p T^{5} + 173 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 96 T^{2} + 6492 T^{4} - 345440 T^{6} + 13937574 T^{8} - 345440 p^{2} T^{10} + 6492 p^{4} T^{12} - 96 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 - 3 T + 57 T^{2} - 195 T^{3} + 1693 T^{4} - 195 p T^{5} + 57 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 279 T^{2} + 36050 T^{4} - 2825121 T^{6} + 147179994 T^{8} - 2825121 p^{2} T^{10} + 36050 p^{4} T^{12} - 279 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 - 166 T^{2} + 15709 T^{4} - 1032577 T^{6} + 53912314 T^{8} - 1032577 p^{2} T^{10} + 15709 p^{4} T^{12} - 166 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 - 292 T^{2} + 40868 T^{4} - 3626556 T^{6} + 226064662 T^{8} - 3626556 p^{2} T^{10} + 40868 p^{4} T^{12} - 292 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 + 11 T + 255 T^{2} + 1896 T^{3} + 23140 T^{4} + 1896 p T^{5} + 255 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 22 T + 372 T^{2} - 4066 T^{3} + 37526 T^{4} - 4066 p T^{5} + 372 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 336 T^{2} + 852 p T^{4} - 6339056 T^{6} + 499727910 T^{8} - 6339056 p^{2} T^{10} + 852 p^{5} T^{12} - 336 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 - 18 T + 351 T^{2} - 3507 T^{3} + 37960 T^{4} - 3507 p T^{5} + 351 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 - 465 T^{2} + 100017 T^{4} - 13107689 T^{6} + 1152750057 T^{8} - 13107689 p^{2} T^{10} + 100017 p^{4} T^{12} - 465 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 + 25 T + 492 T^{2} + 6169 T^{3} + 65294 T^{4} + 6169 p T^{5} + 492 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 463 T^{2} + 101570 T^{4} - 14110809 T^{6} + 1377526138 T^{8} - 14110809 p^{2} T^{10} + 101570 p^{4} T^{12} - 463 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 120 T^{2} - 1848 T^{3} + 1678 T^{4} - 1848 p T^{5} + 120 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( 1 - 272 T^{2} + 49340 T^{4} - 6943920 T^{6} + 739157510 T^{8} - 6943920 p^{2} T^{10} + 49340 p^{4} T^{12} - 272 p^{6} T^{14} + p^{8} T^{16} \)
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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

−3.93087934010871847735702167261, −3.70518358891303163474138699848, −3.64448590660681611224104010985, −3.36497945770705556017326292565, −3.33656660592408159811013639223, −3.13929209735208587478758278026, −2.97622053773548575365803908795, −2.80251862532841403142290329846, −2.79584793983659014345515054551, −2.66765525970923060094549396694, −2.30530935114036861435566496101, −2.29706111638984976526487317633, −2.27338574337178998255145600007, −2.20567850030517188627788734552, −2.14021167683747837110416171225, −1.76392476073156748658557601635, −1.75432982296671438791104751616, −1.37143753572090144994714685428, −1.23740545325966807556889146921, −1.17425981104033382579619072301, −0.976781384595847800173710803267, −0.972767393573252336403373627329, −0.72819592688068853549077723567, −0.62041737772086080669236202684, −0.16392254755592012794825884119, 0.16392254755592012794825884119, 0.62041737772086080669236202684, 0.72819592688068853549077723567, 0.972767393573252336403373627329, 0.976781384595847800173710803267, 1.17425981104033382579619072301, 1.23740545325966807556889146921, 1.37143753572090144994714685428, 1.75432982296671438791104751616, 1.76392476073156748658557601635, 2.14021167683747837110416171225, 2.20567850030517188627788734552, 2.27338574337178998255145600007, 2.29706111638984976526487317633, 2.30530935114036861435566496101, 2.66765525970923060094549396694, 2.79584793983659014345515054551, 2.80251862532841403142290329846, 2.97622053773548575365803908795, 3.13929209735208587478758278026, 3.33656660592408159811013639223, 3.36497945770705556017326292565, 3.64448590660681611224104010985, 3.70518358891303163474138699848, 3.93087934010871847735702167261

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

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