Properties

Label 16-384e8-1.1-c2e8-0-5
Degree $16$
Conductor $4.728\times 10^{20}$
Sign $1$
Analytic cond. $1.43658\times 10^{8}$
Root an. cond. $3.23469$
Motivic weight $2$
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  + 16·5-s − 12·9-s − 48·13-s + 16·17-s + 24·25-s + 80·29-s + 16·37-s + 80·41-s − 192·45-s + 152·49-s − 176·53-s + 272·61-s − 768·65-s − 16·73-s + 90·81-s + 256·85-s − 240·89-s + 400·97-s + 528·101-s − 560·109-s + 336·113-s + 576·117-s + 520·121-s − 1.04e3·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 16/5·5-s − 4/3·9-s − 3.69·13-s + 0.941·17-s + 0.959·25-s + 2.75·29-s + 0.432·37-s + 1.95·41-s − 4.26·45-s + 3.10·49-s − 3.32·53-s + 4.45·61-s − 11.8·65-s − 0.219·73-s + 10/9·81-s + 3.01·85-s − 2.69·89-s + 4.12·97-s + 5.22·101-s − 5.13·109-s + 2.97·113-s + 4.92·117-s + 4.29·121-s − 8.31·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{56} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.43658\times 10^{8}\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{56} \cdot 3^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(10.38724277\)
\(L(\frac12)\) \(\approx\) \(10.38724277\)
\(L(2)\) 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 \( ( 1 + p T^{2} )^{4} \)
good5 \( ( 1 - 8 T + 84 T^{2} - 472 T^{3} + 2822 T^{4} - 472 p^{2} T^{5} + 84 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
7 \( 1 - 152 T^{2} + 1476 p T^{4} - 514600 T^{6} + 25187654 T^{8} - 514600 p^{4} T^{10} + 1476 p^{9} T^{12} - 152 p^{12} T^{14} + p^{16} T^{16} \)
11 \( ( 1 - 260 T^{2} + 40038 T^{4} - 260 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
13 \( ( 1 + 24 T + 540 T^{2} + 5736 T^{3} + 89702 T^{4} + 5736 p^{2} T^{5} + 540 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
17 \( ( 1 - 8 T + 252 T^{2} + 2888 T^{3} + 48902 T^{4} + 2888 p^{2} T^{5} + 252 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
19 \( 1 - 1224 T^{2} + 1027804 T^{4} - 554589432 T^{6} + 236077316358 T^{8} - 554589432 p^{4} T^{10} + 1027804 p^{8} T^{12} - 1224 p^{12} T^{14} + p^{16} T^{16} \)
23 \( 1 - 3272 T^{2} + 5007132 T^{4} - 4715977336 T^{6} + 5669682278 p^{2} T^{8} - 4715977336 p^{4} T^{10} + 5007132 p^{8} T^{12} - 3272 p^{12} T^{14} + p^{16} T^{16} \)
29 \( ( 1 - 40 T + 2196 T^{2} - 31160 T^{3} + 1501766 T^{4} - 31160 p^{2} T^{5} + 2196 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
31 \( 1 - 4248 T^{2} + 9652444 T^{4} - 14917063464 T^{6} + 16677690731718 T^{8} - 14917063464 p^{4} T^{10} + 9652444 p^{8} T^{12} - 4248 p^{12} T^{14} + p^{16} T^{16} \)
37 \( ( 1 - 8 T + 1980 T^{2} + 54920 T^{3} + 1119206 T^{4} + 54920 p^{2} T^{5} + 1980 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
41 \( ( 1 - 40 T + 2556 T^{2} + 43240 T^{3} - 289594 T^{4} + 43240 p^{2} T^{5} + 2556 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
43 \( 1 - 5960 T^{2} + 23041116 T^{4} - 59814611320 T^{6} + 126481252740230 T^{8} - 59814611320 p^{4} T^{10} + 23041116 p^{8} T^{12} - 5960 p^{12} T^{14} + p^{16} T^{16} \)
47 \( 1 - 12360 T^{2} + 70380700 T^{4} - 250403346936 T^{6} + 638353123484742 T^{8} - 250403346936 p^{4} T^{10} + 70380700 p^{8} T^{12} - 12360 p^{12} T^{14} + p^{16} T^{16} \)
53 \( ( 1 + 88 T + 10068 T^{2} + 520520 T^{3} + 37632134 T^{4} + 520520 p^{2} T^{5} + 10068 p^{4} T^{6} + 88 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
59 \( 1 - 7688 T^{2} + 37931484 T^{4} - 124184601784 T^{6} + 382130398823558 T^{8} - 124184601784 p^{4} T^{10} + 37931484 p^{8} T^{12} - 7688 p^{12} T^{14} + p^{16} T^{16} \)
61 \( ( 1 - 136 T + 14076 T^{2} - 1001336 T^{3} + 63819302 T^{4} - 1001336 p^{2} T^{5} + 14076 p^{4} T^{6} - 136 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( ( 1 - 8644 T^{2} + 58097190 T^{4} - 8644 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
71 \( 1 - 18120 T^{2} + 152088604 T^{4} - 903092412024 T^{6} + 4737261672397254 T^{8} - 903092412024 p^{4} T^{10} + 152088604 p^{8} T^{12} - 18120 p^{12} T^{14} + p^{16} T^{16} \)
73 \( ( 1 + 8 T + 5212 T^{2} + 653240 T^{3} - 3523322 T^{4} + 653240 p^{2} T^{5} + 5212 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( 1 - 26776 T^{2} + 333200860 T^{4} - 2815930225960 T^{6} + 19175645821896646 T^{8} - 2815930225960 p^{4} T^{10} + 333200860 p^{8} T^{12} - 26776 p^{12} T^{14} + p^{16} T^{16} \)
83 \( 1 - 15752 T^{2} + 70662492 T^{4} - 102795674680 T^{6} + 590797586843654 T^{8} - 102795674680 p^{4} T^{10} + 70662492 p^{8} T^{12} - 15752 p^{12} T^{14} + p^{16} T^{16} \)
89 \( ( 1 + 120 T + 30300 T^{2} + 2699976 T^{3} + 352873862 T^{4} + 2699976 p^{2} T^{5} + 30300 p^{4} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
97 \( ( 1 - 200 T + 21276 T^{2} - 3009400 T^{3} + 385130822 T^{4} - 3009400 p^{2} T^{5} + 21276 p^{4} T^{6} - 200 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
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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

−4.83861796647265328502114287882, −4.68928606866617477818311187246, −4.62569662611022377245077039767, −4.39224442054104935801887489634, −4.34420048643453464506313510413, −4.06231395579744596012854057051, −3.77239627911664169032447860409, −3.70502391652180962023100693866, −3.64595969747790001748624535247, −3.30780165044552507882672824335, −3.15795273391272359209945141411, −2.79956219368289020129756411452, −2.73781710422507282641429163482, −2.65666432258528588621905580265, −2.51679090231655051418073435117, −2.33479670649663209301427534178, −2.22326937426759364830426984407, −1.98467645101252774206492331805, −1.94349633181754953799497372324, −1.65410291899035953558008272607, −1.52797980866440134138739861454, −0.927063185339134071842689922414, −0.75991007016035743640294325923, −0.53729943089220466036964789454, −0.33762254544829742289622063099, 0.33762254544829742289622063099, 0.53729943089220466036964789454, 0.75991007016035743640294325923, 0.927063185339134071842689922414, 1.52797980866440134138739861454, 1.65410291899035953558008272607, 1.94349633181754953799497372324, 1.98467645101252774206492331805, 2.22326937426759364830426984407, 2.33479670649663209301427534178, 2.51679090231655051418073435117, 2.65666432258528588621905580265, 2.73781710422507282641429163482, 2.79956219368289020129756411452, 3.15795273391272359209945141411, 3.30780165044552507882672824335, 3.64595969747790001748624535247, 3.70502391652180962023100693866, 3.77239627911664169032447860409, 4.06231395579744596012854057051, 4.34420048643453464506313510413, 4.39224442054104935801887489634, 4.62569662611022377245077039767, 4.68928606866617477818311187246, 4.83861796647265328502114287882

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

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