Properties

Label 16-384e8-1.1-c6e8-0-2
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 + 4.46e3·17-s + 1.23e5·25-s − 9.89e4·41-s + 3.31e5·49-s + 7.69e4·73-s + 5.90e5·81-s + 6.63e6·89-s − 4.92e6·97-s − 3.15e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4.33e6·153-s + 157-s + 163-s + 167-s + 1.76e7·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 4/3·9-s + 0.908·17-s + 7.89·25-s − 1.43·41-s + 2.81·49-s + 0.197·73-s + 10/9·81-s + 9.41·89-s − 5.40·97-s − 1.77·121-s + 1.21·153-s + 3.65·169-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

\(L(\frac{7}{2})\) \(\approx\) \(53.27559381\)
\(L(\frac12)\) \(\approx\) \(53.27559381\)
\(L(4)\) 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 \( ( 1 - 1234 p^{2} T^{2} + p^{12} T^{4} )^{4} \)
7 \( ( 1 - 165604 T^{2} + 18091945734 T^{4} - 165604 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
11 \( ( 1 + 1576516 T^{2} + 319487479206 T^{4} + 1576516 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
13 \( ( 1 - 8814820 T^{2} + 39706553357862 T^{4} - 8814820 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
17 \( ( 1 - 1116 T - 2175226 T^{2} - 1116 p^{6} T^{3} + p^{12} T^{4} )^{4} \)
19 \( ( 1 + 154272388 T^{2} + 10374253923756390 T^{4} + 154272388 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
23 \( ( 1 - 279579460 T^{2} + 41996156363862342 T^{4} - 279579460 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
29 \( ( 1 - 3297092 T^{2} + 49459707520391526 T^{4} - 3297092 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
31 \( ( 1 - 2377420708 T^{2} + 2864250173428593990 T^{4} - 2377420708 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
37 \( ( 1 - 4472699044 T^{2} + 17273824012262262246 T^{4} - 4472699044 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
41 \( ( 1 + 24732 T + 9196270886 T^{2} + 24732 p^{6} T^{3} + p^{12} T^{4} )^{4} \)
43 \( ( 1 + 14783863108 T^{2} + \)\(11\!\cdots\!06\)\( T^{4} + 14783863108 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
47 \( ( 1 - 11965054468 T^{2} + 35254807719429905286 T^{4} - 11965054468 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
53 \( ( 1 - 29043574916 T^{2} + \)\(77\!\cdots\!46\)\( T^{4} - 29043574916 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
59 \( ( 1 + 127672933828 T^{2} + \)\(72\!\cdots\!58\)\( T^{4} + 127672933828 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
61 \( ( 1 - 196576229092 T^{2} + \)\(14\!\cdots\!66\)\( T^{4} - 196576229092 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
67 \( ( 1 - 65276716028 T^{2} + \)\(16\!\cdots\!10\)\( T^{4} - 65276716028 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
71 \( ( 1 - 306730507972 T^{2} + \)\(46\!\cdots\!66\)\( T^{4} - 306730507972 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
73 \( ( 1 - 19228 T + 154739682726 T^{2} - 19228 p^{6} T^{3} + p^{12} T^{4} )^{4} \)
79 \( ( 1 - 374724961060 T^{2} + \)\(15\!\cdots\!82\)\( T^{4} - 374724961060 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
83 \( ( 1 + 849400576132 T^{2} + \)\(37\!\cdots\!78\)\( T^{4} + 849400576132 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
89 \( ( 1 - 1659708 T + 1677544070438 T^{2} - 1659708 p^{6} T^{3} + p^{12} T^{4} )^{4} \)
97 \( ( 1 + 1232420 T + 1634488777158 T^{2} + 1232420 p^{6} T^{3} + p^{12} T^{4} )^{4} \)
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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

−4.02946147548982452888783385065, −3.91506123677362789707811725441, −3.44446077065807754147109009133, −3.36504295382554983447618911512, −3.35803496105749866866115515188, −3.21890734905149019689509879670, −3.16389418940789646890479026021, −3.00419726790973500676537927225, −2.68315984098208017332870987040, −2.55709092696575739361760104457, −2.44132941208016368323196081874, −2.39211733015478362630651512981, −2.13833767920018886120978310854, −2.12003606621360491855343777696, −1.62077901557472089359438931819, −1.52985905572160712526736098560, −1.31007729172922154673166727987, −1.26103648517830252407875531868, −1.04988784972774636629409630315, −1.03708052945737353333269523231, −0.986739471808280069230934711453, −0.56523714785408119841472428900, −0.55058827640001653518760982867, −0.38863684617602233642247960013, −0.25292739133215055809237351601, 0.25292739133215055809237351601, 0.38863684617602233642247960013, 0.55058827640001653518760982867, 0.56523714785408119841472428900, 0.986739471808280069230934711453, 1.03708052945737353333269523231, 1.04988784972774636629409630315, 1.26103648517830252407875531868, 1.31007729172922154673166727987, 1.52985905572160712526736098560, 1.62077901557472089359438931819, 2.12003606621360491855343777696, 2.13833767920018886120978310854, 2.39211733015478362630651512981, 2.44132941208016368323196081874, 2.55709092696575739361760104457, 2.68315984098208017332870987040, 3.00419726790973500676537927225, 3.16389418940789646890479026021, 3.21890734905149019689509879670, 3.35803496105749866866115515188, 3.36504295382554983447618911512, 3.44446077065807754147109009133, 3.91506123677362789707811725441, 4.02946147548982452888783385065

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

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