Properties

Label 16-384e8-1.1-c8e8-0-0
Degree $16$
Conductor $4.728\times 10^{20}$
Sign $1$
Analytic cond. $3.58620\times 10^{17}$
Root an. cond. $12.5073$
Motivic weight $8$
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  + 8.74e3·9-s + 9.30e4·17-s + 1.76e6·25-s + 2.97e6·41-s + 3.11e7·49-s + 1.91e8·73-s + 4.78e7·81-s + 4.22e8·89-s + 3.13e7·97-s − 2.27e8·113-s − 4.90e8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8.13e8·153-s + 157-s + 163-s + 167-s + 2.68e9·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 4/3·9-s + 1.11·17-s + 4.50·25-s + 1.05·41-s + 5.40·49-s + 6.73·73-s + 10/9·81-s + 6.73·89-s + 0.354·97-s − 1.39·113-s − 2.28·121-s + 1.48·153-s + 3.28·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+4)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{56} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(3.58620\times 10^{17}\)
Root analytic conductor: \(12.5073\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{56} \cdot 3^{8} ,\ ( \ : [4]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(36.14992703\)
\(L(\frac12)\) \(\approx\) \(36.14992703\)
\(L(5)\) 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 \( ( 1 - p^{7} T^{2} )^{4} \)
good5 \( ( 1 - 880228 T^{2} + 18667806774 p^{2} T^{4} - 880228 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
7 \( ( 1 - 158834 p^{2} T^{2} + p^{16} T^{4} )^{4} \)
11 \( ( 1 + 245275684 T^{2} + 97275433541512902 T^{4} + 245275684 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
13 \( ( 1 - 1340219908 T^{2} + 865655323360267398 T^{4} - 1340219908 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
17 \( ( 1 - 23260 T + 6041535558 T^{2} - 23260 p^{8} T^{3} + p^{16} T^{4} )^{4} \)
19 \( ( 1 + 65391715876 T^{2} + \)\(16\!\cdots\!70\)\( T^{4} + 65391715876 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
23 \( ( 1 - 76045718404 T^{2} + \)\(89\!\cdots\!10\)\( T^{4} - 76045718404 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
29 \( ( 1 - 510589441636 T^{2} + \)\(28\!\cdots\!50\)\( T^{4} - 510589441636 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
31 \( ( 1 - 524615630500 T^{2} + \)\(95\!\cdots\!38\)\( T^{4} - 524615630500 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
37 \( ( 1 - 6548545929604 T^{2} + \)\(27\!\cdots\!42\)\( T^{4} - 6548545929604 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
41 \( ( 1 - 744644 T - 1663452688890 T^{2} - 744644 p^{8} T^{3} + p^{16} T^{4} )^{4} \)
43 \( ( 1 + 34703104689700 T^{2} + \)\(56\!\cdots\!02\)\( T^{4} + 34703104689700 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
47 \( ( 1 - 3500072077444 T^{2} + \)\(10\!\cdots\!10\)\( T^{4} - 3500072077444 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
53 \( ( 1 - 160871699435620 T^{2} + \)\(13\!\cdots\!42\)\( T^{4} - 160871699435620 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
59 \( ( 1 + 197954346294820 T^{2} + \)\(37\!\cdots\!82\)\( T^{4} + 197954346294820 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
61 \( ( 1 - 303812279368324 T^{2} + \)\(49\!\cdots\!22\)\( T^{4} - 303812279368324 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
67 \( ( 1 + 1573471302773284 T^{2} + \)\(94\!\cdots\!62\)\( T^{4} + 1573471302773284 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
71 \( ( 1 - 1674075860991364 T^{2} + \)\(14\!\cdots\!90\)\( T^{4} - 1674075860991364 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
73 \( ( 1 - 47847164 T + 1936048722224262 T^{2} - 47847164 p^{8} T^{3} + p^{16} T^{4} )^{4} \)
79 \( ( 1 - 4210102515025060 T^{2} + \)\(81\!\cdots\!78\)\( T^{4} - 4210102515025060 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
83 \( ( 1 + 7510101349972516 T^{2} + \)\(23\!\cdots\!50\)\( T^{4} + 7510101349972516 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
89 \( ( 1 - 105718108 T + 10045489164843462 T^{2} - 105718108 p^{8} T^{3} + p^{16} T^{4} )^{4} \)
97 \( ( 1 - 7842140 T + 4976272175418438 T^{2} - 7842140 p^{8} T^{3} + p^{16} T^{4} )^{4} \)
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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

−3.65490845555828967169420686169, −3.50263412382311213432350915378, −3.45649288420660646513651124220, −3.16127598827928969080457603607, −3.11520600731085162838610410974, −2.95991202800271838952136413576, −2.84408796788337943569852820654, −2.60275500150708612911232422905, −2.46549691831400983192806043041, −2.26886703675750946904848903658, −2.18388098164911947221144020627, −2.13613369195081718958016151745, −1.98117829297000505904515254093, −1.94794280284226063193648054999, −1.55551913761489378255644078296, −1.27443301092217212962796193534, −1.22586384975650234467532100407, −1.01016221305228105447770612616, −0.938982357159341732681627369970, −0.937539374920302640220203046122, −0.74875471503624393377991841243, −0.70527648633640025926086707533, −0.58205029295005054361915675698, −0.35929757947572135711992583481, −0.13201552623571281877056210231, 0.13201552623571281877056210231, 0.35929757947572135711992583481, 0.58205029295005054361915675698, 0.70527648633640025926086707533, 0.74875471503624393377991841243, 0.937539374920302640220203046122, 0.938982357159341732681627369970, 1.01016221305228105447770612616, 1.22586384975650234467532100407, 1.27443301092217212962796193534, 1.55551913761489378255644078296, 1.94794280284226063193648054999, 1.98117829297000505904515254093, 2.13613369195081718958016151745, 2.18388098164911947221144020627, 2.26886703675750946904848903658, 2.46549691831400983192806043041, 2.60275500150708612911232422905, 2.84408796788337943569852820654, 2.95991202800271838952136413576, 3.11520600731085162838610410974, 3.16127598827928969080457603607, 3.45649288420660646513651124220, 3.50263412382311213432350915378, 3.65490845555828967169420686169

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

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