Properties

Label 16-4032e8-1.1-c1e8-0-13
Degree $16$
Conductor $6.985\times 10^{28}$
Sign $1$
Analytic cond. $1.15446\times 10^{12}$
Root an. cond. $5.67412$
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  + 8·13-s + 4·25-s + 56·37-s − 4·49-s + 16·61-s − 24·73-s − 24·97-s + 96·109-s − 36·121-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 + ⋯
L(s)  = 1  + 2.21·13-s + 4/5·25-s + 9.20·37-s − 4/7·49-s + 2.04·61-s − 2.80·73-s − 2.43·97-s + 9.19·109-s − 3.27·121-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.15446\times 10^{12}\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(27.14817704\)
\(L(\frac12)\) \(\approx\) \(27.14817704\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( ( 1 + T^{2} )^{4} \)
good5 \( ( 1 - 2 T^{2} + 34 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 18 T^{2} + 170 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \)
17 \( ( 1 - 26 T^{2} + 322 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \)
23 \( ( 1 + 66 T^{2} + 1994 T^{4} + 66 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 32 T^{2} + 850 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 20 T^{2} + 934 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 - 14 T + 106 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{4} \)
41 \( ( 1 - 66 T^{2} + 3074 T^{4} - 66 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \)
47 \( ( 1 + 20 T^{2} - 2282 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 128 T^{2} + 8626 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + 132 T^{2} + 8870 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 2 T + p T^{2} )^{8} \)
67 \( ( 1 - 232 T^{2} + 22366 T^{4} - 232 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
71 \( ( 1 + 130 T^{2} + 14154 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \)
79 \( ( 1 - 8 T^{2} + 11886 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 44 T^{2} + 9910 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 338 T^{2} + 44386 T^{4} - 338 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 + 6 T + 186 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \)
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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

−3.38927207561104897307754180923, −3.31656711205520781316859669500, −3.19881341621709906031824415065, −3.06516415540953509468155615314, −2.96722129731254649902363476665, −2.85548694917096672746643027579, −2.85332183506924646546788812807, −2.84302569691666431873180038550, −2.56119245504847395030351656637, −2.28567940056291680613900611608, −2.27197125730796664090847950575, −2.19712465555702103517502153411, −2.15471515085529776824861625493, −2.09076391659608180863808912719, −1.65055234328606156615737896406, −1.56560843308815322639160300195, −1.53416851856952236372750788676, −1.34143856083689557305088106009, −1.01041437354661500351936174155, −1.00897253779843828827818059456, −0.944261313856705002575691191638, −0.803030056834191803262750947423, −0.70914409020238024244004452469, −0.38404535963594481566174694008, −0.30246188345683392244262985063, 0.30246188345683392244262985063, 0.38404535963594481566174694008, 0.70914409020238024244004452469, 0.803030056834191803262750947423, 0.944261313856705002575691191638, 1.00897253779843828827818059456, 1.01041437354661500351936174155, 1.34143856083689557305088106009, 1.53416851856952236372750788676, 1.56560843308815322639160300195, 1.65055234328606156615737896406, 2.09076391659608180863808912719, 2.15471515085529776824861625493, 2.19712465555702103517502153411, 2.27197125730796664090847950575, 2.28567940056291680613900611608, 2.56119245504847395030351656637, 2.84302569691666431873180038550, 2.85332183506924646546788812807, 2.85548694917096672746643027579, 2.96722129731254649902363476665, 3.06516415540953509468155615314, 3.19881341621709906031824415065, 3.31656711205520781316859669500, 3.38927207561104897307754180923

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

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