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Properties

Label 16-3150e8-1.1-c1e8-0-10
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·13-s − 2·16-s + 16·23-s − 16·37-s − 8·43-s − 8·47-s + 32·53-s + 8·59-s − 32·61-s + 16·67-s − 8·83-s − 16·89-s + 16·97-s − 8·103-s − 32·107-s + 32·113-s + 64·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + ⋯
L(s)  = 1  + 2.21·13-s − 1/2·16-s + 3.33·23-s − 2.63·37-s − 1.21·43-s − 1.16·47-s + 4.39·53-s + 1.04·59-s − 4.09·61-s + 1.95·67-s − 0.878·83-s − 1.69·89-s + 1.62·97-s − 0.788·103-s − 3.09·107-s + 3.01·113-s + 5.81·121-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 + 0.0773·167-s + 2.46·169-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

\(L(1)\) \(\approx\) \(4.869602273\)
\(L(\frac12)\) \(\approx\) \(4.869602273\)
\(L(\frac{3}{2})\) 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 \( ( 1 - 32 T^{2} + 466 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( 1 - 8 T + 32 T^{2} - 128 T^{3} + 574 T^{4} - 2360 T^{5} + 8704 T^{6} - 35112 T^{7} + 139491 T^{8} - 35112 p T^{9} + 8704 p^{2} T^{10} - 2360 p^{3} T^{11} + 574 p^{4} T^{12} - 128 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 + 96 T^{3} + 158 T^{4} - 2400 T^{5} + 4608 T^{6} - 8544 T^{7} - 207741 T^{8} - 8544 p T^{9} + 4608 p^{2} T^{10} - 2400 p^{3} T^{11} + 158 p^{4} T^{12} + 96 p^{5} T^{13} + p^{8} T^{16} \)
19 \( ( 1 - 48 T^{2} + 1202 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( 1 - 16 T + 128 T^{2} - 800 T^{3} + 4670 T^{4} - 26576 T^{5} + 147456 T^{6} - 35120 p T^{7} + 7827 p^{2} T^{8} - 35120 p^{2} T^{9} + 147456 p^{2} T^{10} - 26576 p^{3} T^{11} + 4670 p^{4} T^{12} - 800 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
29 \( ( 1 + 2 p T^{2} - 96 T^{3} + 2019 T^{4} - 96 p T^{5} + 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 42 T^{2} - 192 T^{3} + 1139 T^{4} - 192 p T^{5} + 42 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( 1 + 16 T + 128 T^{2} + 784 T^{3} + 4324 T^{4} + 27760 T^{5} + 198016 T^{6} + 1322736 T^{7} + 8422566 T^{8} + 1322736 p T^{9} + 198016 p^{2} T^{10} + 27760 p^{3} T^{11} + 4324 p^{4} T^{12} + 784 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 - 172 T^{2} + 15786 T^{4} - 1010672 T^{6} + 47929043 T^{8} - 1010672 p^{2} T^{10} + 15786 p^{4} T^{12} - 172 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 + 8 T + 32 T^{2} + 368 T^{3} + 1438 T^{4} - 5752 T^{5} - 24320 T^{6} - 270552 T^{7} - 3004701 T^{8} - 270552 p T^{9} - 24320 p^{2} T^{10} - 5752 p^{3} T^{11} + 1438 p^{4} T^{12} + 368 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 + 8 T + 32 T^{2} + 520 T^{3} + 2012 T^{4} - 17432 T^{5} - 68640 T^{6} - 1046296 T^{7} - 15494202 T^{8} - 1046296 p T^{9} - 68640 p^{2} T^{10} - 17432 p^{3} T^{11} + 2012 p^{4} T^{12} + 520 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 32 T + 512 T^{2} - 6592 T^{3} + 79454 T^{4} - 821920 T^{5} + 7348224 T^{6} - 61872608 T^{7} + 480643491 T^{8} - 61872608 p T^{9} + 7348224 p^{2} T^{10} - 821920 p^{3} T^{11} + 79454 p^{4} T^{12} - 6592 p^{5} T^{13} + 512 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \)
59 \( ( 1 - 4 T + 170 T^{2} - 856 T^{3} + 13027 T^{4} - 856 p T^{5} + 170 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 16 T + 222 T^{2} + 2480 T^{3} + 20579 T^{4} + 2480 p T^{5} + 222 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 16 T + 128 T^{2} - 1648 T^{3} + 12196 T^{4} + 9104 T^{5} - 348800 T^{6} + 6630000 T^{7} - 95097114 T^{8} + 6630000 p T^{9} - 348800 p^{2} T^{10} + 9104 p^{3} T^{11} + 12196 p^{4} T^{12} - 1648 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
71 \( ( 1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( 1 + 28 p T^{4} + 20460870 T^{8} + 28 p^{5} T^{12} + p^{8} T^{16} \)
79 \( 1 - 320 T^{2} + 58500 T^{4} - 7347136 T^{6} + 669209222 T^{8} - 7347136 p^{2} T^{10} + 58500 p^{4} T^{12} - 320 p^{6} T^{14} + p^{8} T^{16} \)
83 \( 1 + 8 T + 32 T^{2} + 160 T^{3} + 10142 T^{4} + 122968 T^{5} + 672000 T^{6} + 7632056 T^{7} + 75142083 T^{8} + 7632056 p T^{9} + 672000 p^{2} T^{10} + 122968 p^{3} T^{11} + 10142 p^{4} T^{12} + 160 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
89 \( ( 1 + 8 T + 316 T^{2} + 1912 T^{3} + 40422 T^{4} + 1912 p T^{5} + 316 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 - 16 T + 128 T^{2} - 1136 T^{3} + 10556 T^{4} - 116816 T^{5} + 1163136 T^{6} - 11493168 T^{7} + 113239174 T^{8} - 11493168 p T^{9} + 1163136 p^{2} T^{10} - 116816 p^{3} T^{11} + 10556 p^{4} T^{12} - 1136 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
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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

−3.51889158290718811334210842458, −3.50133597726714942950528179179, −3.41750611450889080586466206106, −3.36796787337537526189161239719, −3.12464286353060354876683231621, −3.08089170336717241536480971263, −2.94003930024234042377523537134, −2.88431323407144779324008155088, −2.55889692384374944975217946092, −2.43807747633239908952576530615, −2.39115011807087687853522091842, −2.29985945266253156283004579131, −2.11454345679736615255568594777, −2.09973434608755725744010237350, −1.86765048066114449724682218052, −1.65112668303865168361135161776, −1.42907564829153804736254489371, −1.36033717764353670725704665031, −1.26838356161347223521965883619, −1.14695769147468203588722410846, −1.12870159742794260796439844728, −0.70406815817534533691427932371, −0.67923771605397600615623007670, −0.34816314193741633628689484018, −0.16804208534377308460233131434, 0.16804208534377308460233131434, 0.34816314193741633628689484018, 0.67923771605397600615623007670, 0.70406815817534533691427932371, 1.12870159742794260796439844728, 1.14695769147468203588722410846, 1.26838356161347223521965883619, 1.36033717764353670725704665031, 1.42907564829153804736254489371, 1.65112668303865168361135161776, 1.86765048066114449724682218052, 2.09973434608755725744010237350, 2.11454345679736615255568594777, 2.29985945266253156283004579131, 2.39115011807087687853522091842, 2.43807747633239908952576530615, 2.55889692384374944975217946092, 2.88431323407144779324008155088, 2.94003930024234042377523537134, 3.08089170336717241536480971263, 3.12464286353060354876683231621, 3.36796787337537526189161239719, 3.41750611450889080586466206106, 3.50133597726714942950528179179, 3.51889158290718811334210842458

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

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