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Properties

Label 16-3150e8-1.1-c1e8-0-22
Degree $16$
Conductor $9.694\times 10^{27}$
Sign $1$
Analytic cond. $1.60214\times 10^{11}$
Root an. cond. $5.01526$
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  + 4·13-s − 2·16-s + 8·23-s + 24·29-s − 8·31-s + 4·37-s + 12·43-s − 12·47-s + 32·53-s + 16·59-s − 20·67-s − 36·73-s + 56·83-s + 72·89-s + 4·97-s + 24·103-s + 16·107-s + 32·113-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 1.10·13-s − 1/2·16-s + 1.66·23-s + 4.45·29-s − 1.43·31-s + 0.657·37-s + 1.82·43-s − 1.75·47-s + 4.39·53-s + 2.08·59-s − 2.44·67-s − 4.21·73-s + 6.14·83-s + 7.63·89-s + 0.406·97-s + 2.36·103-s + 1.54·107-s + 3.01·113-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.60214\times 10^{11}\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(27.87537539\)
\(L(\frac12)\) \(\approx\) \(27.87537539\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7 \( ( 1 + T^{4} )^{2} \)
good11 \( 1 - 12 T^{2} + 40 T^{4} - 260 T^{6} + 15086 T^{8} - 260 p^{2} T^{10} + 40 p^{4} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} \)
13 \( 1 - 4 T + 8 T^{2} - 28 T^{3} + 80 T^{4} - 372 T^{5} + 1240 T^{6} - 6412 T^{7} + 34174 T^{8} - 6412 p T^{9} + 1240 p^{2} T^{10} - 372 p^{3} T^{11} + 80 p^{4} T^{12} - 28 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 + 64 T^{3} - 252 T^{4} - 960 T^{5} + 2048 T^{6} - 6016 T^{7} - 17786 T^{8} - 6016 p T^{9} + 2048 p^{2} T^{10} - 960 p^{3} T^{11} - 252 p^{4} T^{12} + 64 p^{5} T^{13} + p^{8} T^{16} \)
19 \( ( 1 - p T^{2} )^{8} \)
23 \( 1 - 8 T + 32 T^{2} + 8 T^{3} - 924 T^{4} + 5032 T^{5} - 10656 T^{6} - 28456 T^{7} + 420614 T^{8} - 28456 p T^{9} - 10656 p^{2} T^{10} + 5032 p^{3} T^{11} - 924 p^{4} T^{12} + 8 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
29 \( ( 1 - 12 T + 78 T^{2} - 252 T^{3} + 738 T^{4} - 252 p T^{5} + 78 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 4 T + 110 T^{2} + 356 T^{3} + 4930 T^{4} + 356 p T^{5} + 110 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 4 T + 8 T^{2} - 220 T^{3} - 48 T^{4} + 8684 T^{5} - 10152 T^{6} + 208404 T^{7} - 3196994 T^{8} + 208404 p T^{9} - 10152 p^{2} T^{10} + 8684 p^{3} T^{11} - 48 p^{4} T^{12} - 220 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 - 76 T^{2} + 5480 T^{4} - 315940 T^{6} + 12648846 T^{8} - 315940 p^{2} T^{10} + 5480 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 - 12 T + 72 T^{2} - 300 T^{3} - 1008 T^{4} + 17988 T^{5} - 98280 T^{6} + 509316 T^{7} - 2377282 T^{8} + 509316 p T^{9} - 98280 p^{2} T^{10} + 17988 p^{3} T^{11} - 1008 p^{4} T^{12} - 300 p^{5} T^{13} + 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 + 12 T + 72 T^{2} + 220 T^{3} - 2352 T^{4} - 10116 T^{5} + 72152 T^{6} + 1599596 T^{7} + 18271774 T^{8} + 1599596 p T^{9} + 72152 p^{2} T^{10} - 10116 p^{3} T^{11} - 2352 p^{4} T^{12} + 220 p^{5} T^{13} + 72 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 32 T + 512 T^{2} - 5792 T^{3} + 60388 T^{4} - 619872 T^{5} + 5690880 T^{6} - 44817632 T^{7} + 328258854 T^{8} - 44817632 p T^{9} + 5690880 p^{2} T^{10} - 619872 p^{3} T^{11} + 60388 p^{4} T^{12} - 5792 p^{5} T^{13} + 512 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \)
59 \( ( 1 - 8 T + 148 T^{2} - 520 T^{3} + 8710 T^{4} - 520 p T^{5} + 148 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 226 T^{2} + 16 T^{3} + 20162 T^{4} + 16 p T^{5} + 226 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( 1 + 20 T + 200 T^{2} + 1332 T^{3} + 9104 T^{4} + 102596 T^{5} + 1118232 T^{6} + 8938244 T^{7} + 69729150 T^{8} + 8938244 p T^{9} + 1118232 p^{2} T^{10} + 102596 p^{3} T^{11} + 9104 p^{4} T^{12} + 1332 p^{5} T^{13} + 200 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 216 T^{2} + 31388 T^{4} - 3350120 T^{6} + 263319558 T^{8} - 3350120 p^{2} T^{10} + 31388 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 + 36 T + 648 T^{2} + 8620 T^{3} + 98192 T^{4} + 980116 T^{5} + 8807960 T^{6} + 73999132 T^{7} + 618591198 T^{8} + 73999132 p T^{9} + 8807960 p^{2} T^{10} + 980116 p^{3} T^{11} + 98192 p^{4} T^{12} + 8620 p^{5} T^{13} + 648 p^{6} T^{14} + 36 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 - 280 T^{2} + 32732 T^{4} - 1914792 T^{6} + 96372294 T^{8} - 1914792 p^{2} T^{10} + 32732 p^{4} T^{12} - 280 p^{6} T^{14} + p^{8} T^{16} \)
83 \( 1 - 56 T + 1568 T^{2} - 30056 T^{3} + 449924 T^{4} - 5649064 T^{5} + 62548320 T^{6} - 632947128 T^{7} + 5962291750 T^{8} - 632947128 p T^{9} + 62548320 p^{2} T^{10} - 5649064 p^{3} T^{11} + 449924 p^{4} T^{12} - 30056 p^{5} T^{13} + 1568 p^{6} T^{14} - 56 p^{7} T^{15} + p^{8} T^{16} \)
89 \( ( 1 - 36 T + 734 T^{2} - 10004 T^{3} + 106562 T^{4} - 10004 p T^{5} + 734 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 - 4 T + 8 T^{2} - 1852 T^{3} + 7856 T^{4} + 48156 T^{5} + 1459480 T^{6} - 10407580 T^{7} - 90680354 T^{8} - 10407580 p T^{9} + 1459480 p^{2} T^{10} + 48156 p^{3} T^{11} + 7856 p^{4} T^{12} - 1852 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
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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

−3.44706623867051132223828177442, −3.41228755982244574129217258589, −3.33352255449482713431928355951, −3.31902969878940883950723883418, −3.28438097186458471600921194615, −3.05337430997156636288091371999, −2.94636800630101030530047911341, −2.84431194767951064198483114211, −2.66484219710249802420293049454, −2.24079904223674948345022594074, −2.21284176914169607036193782658, −2.20954131877944094732523088243, −2.19520962180976842924003113648, −2.17606727749768082125180184637, −2.16518997249384665978596856208, −1.75261877867588580424357219594, −1.58778492385209691921281190557, −1.13309695910282967825420048890, −1.12520030140201723699817456950, −1.11088966471486031700894447246, −1.03762019743924534077857832254, −0.73100880806499381951908212707, −0.71406542007926834295524987654, −0.45524089846909760924220788309, −0.34724157076307811648912964621, 0.34724157076307811648912964621, 0.45524089846909760924220788309, 0.71406542007926834295524987654, 0.73100880806499381951908212707, 1.03762019743924534077857832254, 1.11088966471486031700894447246, 1.12520030140201723699817456950, 1.13309695910282967825420048890, 1.58778492385209691921281190557, 1.75261877867588580424357219594, 2.16518997249384665978596856208, 2.17606727749768082125180184637, 2.19520962180976842924003113648, 2.20954131877944094732523088243, 2.21284176914169607036193782658, 2.24079904223674948345022594074, 2.66484219710249802420293049454, 2.84431194767951064198483114211, 2.94636800630101030530047911341, 3.05337430997156636288091371999, 3.28438097186458471600921194615, 3.31902969878940883950723883418, 3.33352255449482713431928355951, 3.41228755982244574129217258589, 3.44706623867051132223828177442

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

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