Properties

Degree $16$
Conductor $2.400\times 10^{28}$
Sign $1$
Motivic weight $0$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 10·16-s + 12·23-s − 20·64-s + 81-s − 48·92-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  − 4·4-s + 10·16-s + 12·23-s − 20·64-s + 81-s − 48·92-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 3^{16} \cdot 7^{16}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{3528} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9990037749\)
\(L(\frac12)\) \(\approx\) \(0.9990037749\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{4} \)
3 \( 1 - T^{4} + T^{8} \)
7 \( 1 \)
good5 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
11 \( ( 1 - T^{2} + T^{4} )^{4} \)
13 \( ( 1 - T^{4} + T^{8} )^{2} \)
17 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
19 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
23 \( ( 1 - T )^{8}( 1 - T + T^{2} )^{4} \)
29 \( ( 1 - T^{2} + T^{4} )^{4} \)
31 \( ( 1 + T^{2} )^{8} \)
37 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
41 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
43 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
47 \( ( 1 - T )^{8}( 1 + T )^{8} \)
53 \( ( 1 - T^{2} + T^{4} )^{4} \)
59 \( ( 1 + T^{4} )^{4} \)
61 \( ( 1 - T^{4} + T^{8} )^{2} \)
67 \( ( 1 - T )^{8}( 1 + T )^{8} \)
71 \( ( 1 - T^{2} + T^{4} )^{4} \)
73 \( ( 1 - T^{2} + T^{4} )^{4} \)
79 \( ( 1 - T^{2} + T^{4} )^{4} \)
83 \( ( 1 - T^{4} + T^{8} )^{2} \)
89 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
97 \( ( 1 - T^{2} + T^{4} )^{4} \)
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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

−3.76280385939169173187367722733, −3.50268629685441115190876341914, −3.45514841073596214632302704648, −3.33895036233477295349514455936, −3.23085070703288250942261952430, −3.22372426112599712290023322287, −3.21449256421386131795569253312, −3.08521346697873599964414926152, −3.08286166330424703047554455548, −2.73292651074023791419285258641, −2.72664569764473842005391873648, −2.52433929311177067181008101565, −2.34686414110017071426609793477, −2.31129795111952419094481376649, −2.24748951670814615707599313911, −1.78168855134543336327498769590, −1.54520793365046715443503950040, −1.50218452842280610816521720598, −1.19352398737308935427423036342, −1.16845926179826974997399715363, −1.08569701635602419120883950654, −0.948192522010199699720626754331, −0.907058511574132158144474429475, −0.870459814367495461448571515200, −0.34386840733286357329229250223, 0.34386840733286357329229250223, 0.870459814367495461448571515200, 0.907058511574132158144474429475, 0.948192522010199699720626754331, 1.08569701635602419120883950654, 1.16845926179826974997399715363, 1.19352398737308935427423036342, 1.50218452842280610816521720598, 1.54520793365046715443503950040, 1.78168855134543336327498769590, 2.24748951670814615707599313911, 2.31129795111952419094481376649, 2.34686414110017071426609793477, 2.52433929311177067181008101565, 2.72664569764473842005391873648, 2.73292651074023791419285258641, 3.08286166330424703047554455548, 3.08521346697873599964414926152, 3.21449256421386131795569253312, 3.22372426112599712290023322287, 3.23085070703288250942261952430, 3.33895036233477295349514455936, 3.45514841073596214632302704648, 3.50268629685441115190876341914, 3.76280385939169173187367722733

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

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