Properties

Label 16-384e8-1.1-c6e8-0-0
Degree $16$
Conductor $4.728\times 10^{20}$
Sign $1$
Analytic cond. $3.70927\times 10^{15}$
Root an. cond. $9.39897$
Motivic weight $6$
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  + 972·9-s − 6.54e3·17-s + 3.41e4·25-s − 4.99e5·41-s + 2.63e5·49-s − 1.96e6·73-s + 5.90e5·81-s − 1.69e6·89-s + 7.63e6·97-s + 9.44e6·113-s − 5.08e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 6.36e6·153-s + 157-s + 163-s + 167-s + 2.04e7·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 4/3·9-s − 1.33·17-s + 2.18·25-s − 7.24·41-s + 2.23·49-s − 5.04·73-s + 10/9·81-s − 2.40·89-s + 8.36·97-s + 6.54·113-s − 2.86·121-s − 1.77·153-s + 4.23·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{56} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(3.70927\times 10^{15}\)
Root analytic conductor: \(9.39897\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{56} \cdot 3^{8} ,\ ( \ : [3]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.952004520\)
\(L(\frac12)\) \(\approx\) \(3.952004520\)
\(L(4)\) 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 \( ( 1 - p^{5} T^{2} )^{4} \)
good5 \( ( 1 - 17092 T^{2} + 518022 p^{4} T^{4} - 17092 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
7 \( ( 1 - 131620 T^{2} + 23461672326 T^{4} - 131620 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
11 \( ( 1 + 2541124 T^{2} + 7248844979622 T^{4} + 2541124 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
13 \( ( 1 - 10218340 T^{2} + 63759295714086 T^{4} - 10218340 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
17 \( ( 1 + 1636 T - 10443642 T^{2} + 1636 p^{6} T^{3} + p^{12} T^{4} )^{4} \)
19 \( ( 1 + 60040324 T^{2} + 3330030874078566 T^{4} + 60040324 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
23 \( ( 1 - 411628612 T^{2} + 81039053669311302 T^{4} - 411628612 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
29 \( ( 1 - 509438980 T^{2} + 21871908390235878 T^{4} - 509438980 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
31 \( ( 1 + 1726530332 T^{2} + 1819945342319196102 T^{4} + 1726530332 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
37 \( ( 1 - 3141266212 T^{2} + 3988917501941351142 T^{4} - 3141266212 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
41 \( ( 1 + 124892 T + 13340323494 T^{2} + 124892 p^{6} T^{3} + p^{12} T^{4} )^{4} \)
43 \( ( 1 + 22395690820 T^{2} + \)\(20\!\cdots\!02\)\( T^{4} + 22395690820 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
47 \( ( 1 - 16459395844 T^{2} + \)\(20\!\cdots\!22\)\( T^{4} - 16459395844 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
53 \( ( 1 - 2028435268 T^{2} + 24452922367168138662 T^{4} - 2028435268 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
59 \( ( 1 + 1852441868 p T^{2} + \)\(62\!\cdots\!22\)\( T^{4} + 1852441868 p^{13} T^{6} + p^{24} T^{8} )^{2} \)
61 \( ( 1 - 196203900772 T^{2} + \)\(14\!\cdots\!42\)\( T^{4} - 196203900772 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
67 \( ( 1 + 220852912132 T^{2} + \)\(23\!\cdots\!78\)\( T^{4} + 220852912132 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
71 \( ( 1 + 57193478204 T^{2} + \)\(76\!\cdots\!42\)\( T^{4} + 57193478204 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
73 \( ( 1 + 490660 T + 339100099878 T^{2} + 490660 p^{6} T^{3} + p^{12} T^{4} )^{4} \)
79 \( ( 1 - 181850004580 T^{2} + \)\(84\!\cdots\!78\)\( T^{4} - 181850004580 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
83 \( ( 1 + 798104274052 T^{2} + \)\(33\!\cdots\!98\)\( T^{4} + 798104274052 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
89 \( ( 1 + 423748 T + 953097040422 T^{2} + 423748 p^{6} T^{3} + p^{12} T^{4} )^{4} \)
97 \( ( 1 - 1908188 T + 2570300580294 T^{2} - 1908188 p^{6} T^{3} + p^{12} T^{4} )^{4} \)
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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

−3.96038842548633975967313494233, −3.82831687435601877280185219003, −3.51444542148108119948737122130, −3.47409445591221360336973023396, −3.46186948844699706787561588811, −3.32261477761151924592233971145, −2.96632240982487670020575703611, −2.91985305847968993221634815183, −2.79900981900025898764319447287, −2.68338987265155135965706601989, −2.38105424716477628617959250810, −2.31224649468550840717115733339, −2.00621709753315856941753238121, −1.75146378184646070197170350343, −1.73314720480577978607540330196, −1.71262745565371291798896467002, −1.59692100893727783528057529757, −1.42473800533370296503052305837, −1.19550638133090670537848892657, −0.879527115111115201654765081076, −0.68351135096152561464569701172, −0.61392696382008788483802281054, −0.44615790281898027238989922607, −0.39642024436225865404325638332, −0.088891262865096915654709499252, 0.088891262865096915654709499252, 0.39642024436225865404325638332, 0.44615790281898027238989922607, 0.61392696382008788483802281054, 0.68351135096152561464569701172, 0.879527115111115201654765081076, 1.19550638133090670537848892657, 1.42473800533370296503052305837, 1.59692100893727783528057529757, 1.71262745565371291798896467002, 1.73314720480577978607540330196, 1.75146378184646070197170350343, 2.00621709753315856941753238121, 2.31224649468550840717115733339, 2.38105424716477628617959250810, 2.68338987265155135965706601989, 2.79900981900025898764319447287, 2.91985305847968993221634815183, 2.96632240982487670020575703611, 3.32261477761151924592233971145, 3.46186948844699706787561588811, 3.47409445591221360336973023396, 3.51444542148108119948737122130, 3.82831687435601877280185219003, 3.96038842548633975967313494233

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

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