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Properties

Label 16-48e8-1.1-c13e8-0-0
Degree $16$
Conductor $2.818\times 10^{13}$
Sign $1$
Analytic cond. $4.92595\times 10^{13}$
Root an. cond. $7.17431$
Motivic weight $13$
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  − 1.38e6·9-s − 1.66e7·13-s + 3.03e9·25-s + 2.97e10·37-s + 4.01e11·49-s − 1.08e12·61-s + 3.40e12·73-s + 2.22e12·81-s − 1.05e13·97-s − 5.02e13·109-s + 2.29e13·117-s − 1.47e14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.05e15·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 0.868·9-s − 0.954·13-s + 2.48·25-s + 1.90·37-s + 4.14·49-s − 2.68·61-s + 2.63·73-s + 0.876·81-s − 1.28·97-s − 2.86·109-s + 0.828·117-s − 4.25·121-s − 6.78·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(14-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+13/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(4.92595\times 10^{13}\)
Root analytic conductor: \(7.17431\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 3^{8} ,\ ( \ : [13/2]^{8} ),\ 1 )\)

Particular Values

\(L(7)\) \(\approx\) \(3.185323780\)
\(L(\frac12)\) \(\approx\) \(3.185323780\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + 51268 p^{3} T^{2} - 194894 p^{13} T^{4} + 51268 p^{29} T^{6} + p^{52} T^{8} \)
good5 \( ( 1 - 1515117044 T^{2} + 7403081934254286 p^{3} T^{4} - 1515117044 p^{26} T^{6} + p^{52} T^{8} )^{2} \)
7 \( ( 1 - 200563962724 T^{2} + \)\(58\!\cdots\!54\)\( p^{2} T^{4} - 200563962724 p^{26} T^{6} + p^{52} T^{8} )^{2} \)
11 \( ( 1 + 6682268588260 p T^{2} + \)\(20\!\cdots\!58\)\( p^{3} T^{4} + 6682268588260 p^{27} T^{6} + p^{52} T^{8} )^{2} \)
13 \( ( 1 + 319444 p T + 557200245281406 T^{2} + 319444 p^{14} T^{3} + p^{26} T^{4} )^{4} \)
17 \( ( 1 - 23694243851937092 T^{2} + \)\(29\!\cdots\!70\)\( T^{4} - 23694243851937092 p^{26} T^{6} + p^{52} T^{8} )^{2} \)
19 \( ( 1 - 151903738754763412 T^{2} + \)\(93\!\cdots\!54\)\( T^{4} - 151903738754763412 p^{26} T^{6} + p^{52} T^{8} )^{2} \)
23 \( ( 1 + 1208194792047327836 T^{2} + \)\(71\!\cdots\!58\)\( T^{4} + 1208194792047327836 p^{26} T^{6} + p^{52} T^{8} )^{2} \)
29 \( ( 1 - 20730661708717646036 T^{2} + \)\(29\!\cdots\!66\)\( T^{4} - 20730661708717646036 p^{26} T^{6} + p^{52} T^{8} )^{2} \)
31 \( ( 1 - 56016984024910009348 T^{2} + \)\(18\!\cdots\!74\)\( T^{4} - 56016984024910009348 p^{26} T^{6} + p^{52} T^{8} )^{2} \)
37 \( ( 1 - 7449024172 T + \)\(32\!\cdots\!94\)\( T^{2} - 7449024172 p^{13} T^{3} + p^{26} T^{4} )^{4} \)
41 \( ( 1 + \)\(40\!\cdots\!92\)\( T^{2} + \)\(15\!\cdots\!34\)\( T^{4} + \)\(40\!\cdots\!92\)\( p^{26} T^{6} + p^{52} T^{8} )^{2} \)
43 \( ( 1 - \)\(51\!\cdots\!24\)\( T^{2} + \)\(11\!\cdots\!78\)\( T^{4} - \)\(51\!\cdots\!24\)\( p^{26} T^{6} + p^{52} T^{8} )^{2} \)
47 \( ( 1 + \)\(14\!\cdots\!48\)\( T^{2} + \)\(44\!\cdots\!34\)\( T^{4} + \)\(14\!\cdots\!48\)\( p^{26} T^{6} + p^{52} T^{8} )^{2} \)
53 \( ( 1 - \)\(55\!\cdots\!96\)\( T^{2} + \)\(21\!\cdots\!58\)\( T^{4} - \)\(55\!\cdots\!96\)\( p^{26} T^{6} + p^{52} T^{8} )^{2} \)
59 \( ( 1 - \)\(19\!\cdots\!20\)\( T^{2} + \)\(25\!\cdots\!58\)\( T^{4} - \)\(19\!\cdots\!20\)\( p^{26} T^{6} + p^{52} T^{8} )^{2} \)
61 \( ( 1 + 270375011300 T + \)\(13\!\cdots\!58\)\( T^{2} + 270375011300 p^{13} T^{3} + p^{26} T^{4} )^{4} \)
67 \( ( 1 + \)\(34\!\cdots\!64\)\( T^{2} + \)\(62\!\cdots\!98\)\( T^{4} + \)\(34\!\cdots\!64\)\( p^{26} T^{6} + p^{52} T^{8} )^{2} \)
71 \( ( 1 - \)\(41\!\cdots\!60\)\( T^{2} + \)\(81\!\cdots\!38\)\( T^{4} - \)\(41\!\cdots\!60\)\( p^{26} T^{6} + p^{52} T^{8} )^{2} \)
73 \( ( 1 - 851945785556 T + \)\(77\!\cdots\!66\)\( T^{2} - 851945785556 p^{13} T^{3} + p^{26} T^{4} )^{4} \)
79 \( ( 1 - \)\(40\!\cdots\!32\)\( T^{2} + \)\(42\!\cdots\!54\)\( T^{4} - \)\(40\!\cdots\!32\)\( p^{26} T^{6} + p^{52} T^{8} )^{2} \)
83 \( ( 1 + \)\(22\!\cdots\!08\)\( T^{2} + \)\(25\!\cdots\!10\)\( T^{4} + \)\(22\!\cdots\!08\)\( p^{26} T^{6} + p^{52} T^{8} )^{2} \)
89 \( ( 1 - \)\(78\!\cdots\!72\)\( T^{2} + \)\(25\!\cdots\!34\)\( T^{4} - \)\(78\!\cdots\!72\)\( p^{26} T^{6} + p^{52} T^{8} )^{2} \)
97 \( ( 1 + 2630114237500 T + \)\(13\!\cdots\!54\)\( T^{2} + 2630114237500 p^{13} T^{3} + p^{26} T^{4} )^{4} \)
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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

−4.83342642204964626564156955823, −4.67361768745370295645469777031, −4.56371696243192609719007959688, −3.88489389721455918400409015903, −3.86847383102220066424404316456, −3.73799939531277305987202916231, −3.70924375501307718469573537607, −3.64131845004778209130067478434, −3.21334136931925674981104430093, −2.81308111212888582186139901250, −2.65840399648063032736643048261, −2.52924777785885719472008743522, −2.47970642809623692815093178796, −2.47357204030215494852240001847, −2.42459803184691721105476019904, −1.98596308188612334900040722218, −1.37469828734082369034614898373, −1.34785565783493425726349972756, −1.28535947717839362452676522798, −1.24419526304318399451710832792, −0.945616574875663227663157438347, −0.75006471897811514848663455723, −0.34418459341656028151469381927, −0.30639001640785355618097647806, −0.16076185496219949885191687197, 0.16076185496219949885191687197, 0.30639001640785355618097647806, 0.34418459341656028151469381927, 0.75006471897811514848663455723, 0.945616574875663227663157438347, 1.24419526304318399451710832792, 1.28535947717839362452676522798, 1.34785565783493425726349972756, 1.37469828734082369034614898373, 1.98596308188612334900040722218, 2.42459803184691721105476019904, 2.47357204030215494852240001847, 2.47970642809623692815093178796, 2.52924777785885719472008743522, 2.65840399648063032736643048261, 2.81308111212888582186139901250, 3.21334136931925674981104430093, 3.64131845004778209130067478434, 3.70924375501307718469573537607, 3.73799939531277305987202916231, 3.86847383102220066424404316456, 3.88489389721455918400409015903, 4.56371696243192609719007959688, 4.67361768745370295645469777031, 4.83342642204964626564156955823

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

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