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Properties

Label 16-930e8-1.1-c1e8-0-3
Degree $16$
Conductor $5.596\times 10^{23}$
Sign $1$
Analytic cond. $9.24869\times 10^{6}$
Root an. cond. $2.72508$
Motivic weight $1$
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  + 8·3-s + 4·7-s + 32·9-s − 4·13-s − 2·16-s − 4·17-s + 32·21-s + 12·23-s − 2·25-s + 88·27-s + 8·29-s + 8·31-s − 16·37-s − 32·39-s − 8·43-s − 28·47-s − 16·48-s + 8·49-s − 32·51-s + 12·53-s − 24·59-s + 56·61-s + 128·63-s − 16·67-s + 96·69-s − 36·73-s − 16·75-s + ⋯
L(s)  = 1  + 4.61·3-s + 1.51·7-s + 32/3·9-s − 1.10·13-s − 1/2·16-s − 0.970·17-s + 6.98·21-s + 2.50·23-s − 2/5·25-s + 16.9·27-s + 1.48·29-s + 1.43·31-s − 2.63·37-s − 5.12·39-s − 1.21·43-s − 4.08·47-s − 2.30·48-s + 8/7·49-s − 4.48·51-s + 1.64·53-s − 3.12·59-s + 7.17·61-s + 16.1·63-s − 1.95·67-s + 11.5·69-s − 4.21·73-s − 1.84·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 31^{8}\)
Sign: $1$
Analytic conductor: \(9.24869\times 10^{6}\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 31^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(18.67822576\)
\(L(\frac12)\) \(\approx\) \(18.67822576\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{4} )^{2} \)
3 \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
5 \( 1 + 2 T^{2} - 16 T^{3} + 2 T^{4} - 16 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \)
31 \( ( 1 - T )^{8} \)
good7 \( 1 - 4 T + 8 T^{2} + 20 T^{3} + 5 T^{4} - 312 T^{5} + 1408 T^{6} - 64 T^{7} - 3356 T^{8} - 64 p T^{9} + 1408 p^{2} T^{10} - 312 p^{3} T^{11} + 5 p^{4} T^{12} + 20 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
11 \( 1 - 6 p T^{2} + 2033 T^{4} - 38730 T^{6} + 506548 T^{8} - 38730 p^{2} T^{10} + 2033 p^{4} T^{12} - 6 p^{7} T^{14} + p^{8} T^{16} \)
13 \( 1 + 4 T + 8 T^{2} + 76 T^{3} + 128 T^{4} - 796 T^{5} - 1320 T^{6} - 11668 T^{7} - 98082 T^{8} - 11668 p T^{9} - 1320 p^{2} T^{10} - 796 p^{3} T^{11} + 128 p^{4} T^{12} + 76 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 + 4 T + 8 T^{2} - 100 T^{3} - 1152 T^{4} - 3932 T^{5} - 1512 T^{6} + 69692 T^{7} + 524414 T^{8} + 69692 p T^{9} - 1512 p^{2} T^{10} - 3932 p^{3} T^{11} - 1152 p^{4} T^{12} - 100 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
19 \( 1 - 34 T^{2} + 1073 T^{4} - 27050 T^{6} + 566548 T^{8} - 27050 p^{2} T^{10} + 1073 p^{4} T^{12} - 34 p^{6} T^{14} + p^{8} T^{16} \)
23 \( 1 - 12 T + 72 T^{2} - 412 T^{3} + 83 p T^{4} - 5800 T^{5} + 17024 T^{6} - 33848 T^{7} - 19404 T^{8} - 33848 p T^{9} + 17024 p^{2} T^{10} - 5800 p^{3} T^{11} + 83 p^{5} T^{12} - 412 p^{5} T^{13} + 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
29 \( ( 1 - 4 T + 54 T^{2} - 356 T^{3} + 1698 T^{4} - 356 p T^{5} + 54 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \)
41 \( ( 1 - 4 T - 58 T^{2} + 60 T^{3} + 3074 T^{4} + 60 p T^{5} - 58 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )( 1 + 4 T - 58 T^{2} - 60 T^{3} + 3074 T^{4} - 60 p T^{5} - 58 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} ) \)
43 \( 1 + 8 T + 32 T^{2} + 240 T^{3} + 1817 T^{4} + 11648 T^{5} + 63840 T^{6} + 654296 T^{7} + 6747168 T^{8} + 654296 p T^{9} + 63840 p^{2} T^{10} + 11648 p^{3} T^{11} + 1817 p^{4} T^{12} + 240 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 + 28 T + 392 T^{2} + 3932 T^{3} + 37120 T^{4} + 348700 T^{5} + 2942872 T^{6} + 21477468 T^{7} + 147370878 T^{8} + 21477468 p T^{9} + 2942872 p^{2} T^{10} + 348700 p^{3} T^{11} + 37120 p^{4} T^{12} + 3932 p^{5} T^{13} + 392 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 12 T + 72 T^{2} + 136 T^{3} - 167 T^{4} + 256 T^{5} + 18200 T^{6} + 1942052 T^{7} - 14153856 T^{8} + 1942052 p T^{9} + 18200 p^{2} T^{10} + 256 p^{3} T^{11} - 167 p^{4} T^{12} + 136 p^{5} T^{13} + 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
59 \( ( 1 + 12 T + 206 T^{2} + 1612 T^{3} + 16322 T^{4} + 1612 p T^{5} + 206 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 28 T + 430 T^{2} - 4404 T^{3} + 37226 T^{4} - 4404 p T^{5} + 430 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 + 16 T + 128 T^{2} + 624 T^{3} - 988 T^{4} - 12176 T^{5} + 126336 T^{6} + 2338192 T^{7} + 28104678 T^{8} + 2338192 p T^{9} + 126336 p^{2} T^{10} - 12176 p^{3} T^{11} - 988 p^{4} T^{12} + 624 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 282 T^{2} + 41873 T^{4} - 4454666 T^{6} + 362021124 T^{8} - 4454666 p^{2} T^{10} + 41873 p^{4} T^{12} - 282 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 + 36 T + 648 T^{2} + 9104 T^{3} + 117257 T^{4} + 1342448 T^{5} + 13787000 T^{6} + 134690396 T^{7} + 1220953488 T^{8} + 134690396 p T^{9} + 13787000 p^{2} T^{10} + 1342448 p^{3} T^{11} + 117257 p^{4} T^{12} + 9104 p^{5} T^{13} + 648 p^{6} T^{14} + 36 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 - 354 T^{2} + 63265 T^{4} - 7650250 T^{6} + 690839396 T^{8} - 7650250 p^{2} T^{10} + 63265 p^{4} T^{12} - 354 p^{6} T^{14} + p^{8} T^{16} \)
83 \( 1 - 12 T + 72 T^{2} - 1884 T^{3} + 11376 T^{4} + 25716 T^{5} + 647064 T^{6} - 1901724 T^{7} - 83486146 T^{8} - 1901724 p T^{9} + 647064 p^{2} T^{10} + 25716 p^{3} T^{11} + 11376 p^{4} T^{12} - 1884 p^{5} T^{13} + 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
89 \( ( 1 + 18 T + 469 T^{2} + 5094 T^{3} + 67888 T^{4} + 5094 p T^{5} + 469 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 - 12 T + 72 T^{2} - 820 T^{3} + 1712 T^{4} + 8468 T^{5} + 111320 T^{6} - 3224596 T^{7} + 72159198 T^{8} - 3224596 p T^{9} + 111320 p^{2} T^{10} + 8468 p^{3} T^{11} + 1712 p^{4} T^{12} - 820 p^{5} T^{13} + 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
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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

−4.27968718251276563259109607665, −4.26353740701282673210352689146, −4.01715961208425156770059332855, −3.70170663543534419359820401990, −3.62630719694428483288209317294, −3.49367260250058818413397624687, −3.47679101688059659460507175894, −3.36492810591506052887098453050, −3.36262334299914255444668725623, −2.95524078843954083905238207636, −2.94639910168199568264377240705, −2.67463820966066200994088690898, −2.66795919888763185612535103434, −2.60991063663461614211216008467, −2.49357585876319007795012210517, −2.33251116445697429416625424523, −2.02173657495387572995264952649, −1.88109378179421902557745172717, −1.84143773934817705429421597303, −1.65569900323270649212755769577, −1.39324981395218617842710575562, −1.38852571434403640442505073796, −0.894098459272542936382938132501, −0.884386930885133099616973180904, −0.20285975959512310302288593211, 0.20285975959512310302288593211, 0.884386930885133099616973180904, 0.894098459272542936382938132501, 1.38852571434403640442505073796, 1.39324981395218617842710575562, 1.65569900323270649212755769577, 1.84143773934817705429421597303, 1.88109378179421902557745172717, 2.02173657495387572995264952649, 2.33251116445697429416625424523, 2.49357585876319007795012210517, 2.60991063663461614211216008467, 2.66795919888763185612535103434, 2.67463820966066200994088690898, 2.94639910168199568264377240705, 2.95524078843954083905238207636, 3.36262334299914255444668725623, 3.36492810591506052887098453050, 3.47679101688059659460507175894, 3.49367260250058818413397624687, 3.62630719694428483288209317294, 3.70170663543534419359820401990, 4.01715961208425156770059332855, 4.26353740701282673210352689146, 4.27968718251276563259109607665

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

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