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Properties

Label 16-4056e8-1.1-c1e8-0-0
Degree $16$
Conductor $7.325\times 10^{28}$
Sign $1$
Analytic cond. $1.21059\times 10^{12}$
Root an. cond. $5.69098$
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 + 36·9-s − 24·17-s + 4·23-s + 18·25-s + 120·27-s − 12·29-s + 16·43-s + 10·49-s − 192·51-s + 4·53-s − 4·61-s + 32·69-s + 144·75-s − 12·79-s + 330·81-s − 96·87-s + 12·101-s − 32·103-s − 44·107-s + 24·113-s + 24·121-s + 127-s + 128·129-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 4.61·3-s + 12·9-s − 5.82·17-s + 0.834·23-s + 18/5·25-s + 23.0·27-s − 2.22·29-s + 2.43·43-s + 10/7·49-s − 26.8·51-s + 0.549·53-s − 0.512·61-s + 3.85·69-s + 16.6·75-s − 1.35·79-s + 36.6·81-s − 10.2·87-s + 1.19·101-s − 3.15·103-s − 4.25·107-s + 2.25·113-s + 2.18·121-s + 0.0887·127-s + 11.2·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 3^{8} \cdot 13^{16}\)
Sign: $1$
Analytic conductor: \(1.21059\times 10^{12}\)
Root analytic conductor: \(5.69098\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 3^{8} \cdot 13^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(56.35110674\)
\(L(\frac12)\) \(\approx\) \(56.35110674\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 - T )^{8} \)
13 \( 1 \)
good5 \( 1 - 18 T^{2} + 121 T^{4} - 282 T^{6} + 36 T^{8} - 282 p^{2} T^{10} + 121 p^{4} T^{12} - 18 p^{6} T^{14} + p^{8} T^{16} \)
7 \( 1 - 10 T^{2} + 157 T^{4} - 1042 T^{6} + 10096 T^{8} - 1042 p^{2} T^{10} + 157 p^{4} T^{12} - 10 p^{6} T^{14} + p^{8} T^{16} \)
11 \( 1 - 24 T^{2} + 400 T^{4} - 5880 T^{6} + 77982 T^{8} - 5880 p^{2} T^{10} + 400 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \)
17 \( ( 1 + 12 T + 101 T^{2} + 36 p T^{3} + 2808 T^{4} + 36 p^{2} T^{5} + 101 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( 1 - 40 T^{2} + 976 T^{4} - 13288 T^{6} + 229726 T^{8} - 13288 p^{2} T^{10} + 976 p^{4} T^{12} - 40 p^{6} T^{14} + p^{8} T^{16} \)
23 \( ( 1 - 2 T + 62 T^{2} - 170 T^{3} + 1786 T^{4} - 170 p T^{5} + 62 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + 6 T + 101 T^{2} + 486 T^{3} + 4212 T^{4} + 486 p T^{5} + 101 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( 1 - 130 T^{2} + 9433 T^{4} - 464794 T^{6} + 16726996 T^{8} - 464794 p^{2} T^{10} + 9433 p^{4} T^{12} - 130 p^{6} T^{14} + p^{8} T^{16} \)
37 \( 1 - 214 T^{2} + 21553 T^{4} - 1354894 T^{6} + 59197588 T^{8} - 1354894 p^{2} T^{10} + 21553 p^{4} T^{12} - 214 p^{6} T^{14} + p^{8} T^{16} \)
41 \( 1 - 118 T^{2} + 9217 T^{4} - 472270 T^{6} + 21506596 T^{8} - 472270 p^{2} T^{10} + 9217 p^{4} T^{12} - 118 p^{6} T^{14} + p^{8} T^{16} \)
43 \( ( 1 - 8 T + 127 T^{2} - 842 T^{3} + 7918 T^{4} - 842 p T^{5} + 127 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 - 216 T^{2} + 26032 T^{4} - 2028984 T^{6} + 112903998 T^{8} - 2028984 p^{2} T^{10} + 26032 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16} \)
53 \( ( 1 - 2 T + 65 T^{2} - 566 T^{3} + 1480 T^{4} - 566 p T^{5} + 65 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 - 288 T^{2} + 42076 T^{4} - 4053984 T^{6} + 279762726 T^{8} - 4053984 p^{2} T^{10} + 42076 p^{4} T^{12} - 288 p^{6} T^{14} + p^{8} T^{16} \)
61 \( ( 1 + 2 T + 46 T^{2} - 64 T^{3} + 4651 T^{4} - 64 p T^{5} + 46 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 346 T^{2} + 53005 T^{4} - 5015698 T^{6} + 364096192 T^{8} - 5015698 p^{2} T^{10} + 53005 p^{4} T^{12} - 346 p^{6} T^{14} + p^{8} T^{16} \)
71 \( 1 - 136 T^{2} + 5872 T^{4} - 702472 T^{6} + 88350334 T^{8} - 702472 p^{2} T^{10} + 5872 p^{4} T^{12} - 136 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 - 460 T^{2} + 99418 T^{4} - 13139728 T^{6} + 1160088163 T^{8} - 13139728 p^{2} T^{10} + 99418 p^{4} T^{12} - 460 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 + 6 T + 253 T^{2} + 1314 T^{3} + 27708 T^{4} + 1314 p T^{5} + 253 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 352 T^{2} + 60208 T^{4} - 6828304 T^{6} + 613131550 T^{8} - 6828304 p^{2} T^{10} + 60208 p^{4} T^{12} - 352 p^{6} T^{14} + p^{8} T^{16} \)
89 \( 1 - 232 T^{2} + 33628 T^{4} - 4279000 T^{6} + 431138182 T^{8} - 4279000 p^{2} T^{10} + 33628 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \)
97 \( 1 - 362 T^{2} + 78433 T^{4} - 11635826 T^{6} + 1298154244 T^{8} - 11635826 p^{2} T^{10} + 78433 p^{4} T^{12} - 362 p^{6} T^{14} + p^{8} T^{16} \)
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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

−3.54331885276981336444897037057, −3.07971140870810128830133016271, −2.99947826157638680585515349147, −2.97369522312705211602280080379, −2.95857097636597015222683276335, −2.93375583494545520557730594951, −2.89428897541568845190405026270, −2.78410752033627454538115634491, −2.53886254645834849219701867316, −2.51858643019396176845908284705, −2.28968905733166634437059105252, −2.09277952694145078016116976054, −2.05929467770813405447883878920, −2.05459677083953340158483674486, −1.91918039504009686041051121014, −1.87350540800299602438473170597, −1.72573455496222565240452253598, −1.48842128451099523404396681969, −1.44814282174146379196212477250, −0.990151024950199176569300151902, −0.919116616979478201768076090094, −0.868042074206528444150684491004, −0.794229663308374786265739739641, −0.29793656963070699983934098918, −0.24923933183936493395387143441, 0.24923933183936493395387143441, 0.29793656963070699983934098918, 0.794229663308374786265739739641, 0.868042074206528444150684491004, 0.919116616979478201768076090094, 0.990151024950199176569300151902, 1.44814282174146379196212477250, 1.48842128451099523404396681969, 1.72573455496222565240452253598, 1.87350540800299602438473170597, 1.91918039504009686041051121014, 2.05459677083953340158483674486, 2.05929467770813405447883878920, 2.09277952694145078016116976054, 2.28968905733166634437059105252, 2.51858643019396176845908284705, 2.53886254645834849219701867316, 2.78410752033627454538115634491, 2.89428897541568845190405026270, 2.93375583494545520557730594951, 2.95857097636597015222683276335, 2.97369522312705211602280080379, 2.99947826157638680585515349147, 3.07971140870810128830133016271, 3.54331885276981336444897037057

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

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