Properties

Label 16-429e8-1.1-c1e8-0-1
Degree $16$
Conductor $1.147\times 10^{21}$
Sign $1$
Analytic cond. $18961.6$
Root an. cond. $1.85083$
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  + 12·4-s − 12·9-s + 74·16-s − 144·36-s − 8·49-s + 300·64-s + 90·81-s − 64·103-s + 36·121-s + 127-s + 131-s + 137-s + 139-s − 888·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 96·196-s + 197-s + ⋯
L(s)  = 1  + 6·4-s − 4·9-s + 37/2·16-s − 24·36-s − 8/7·49-s + 75/2·64-s + 10·81-s − 6.30·103-s + 3.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 74·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 6.85·196-s + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 11^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(18961.6\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 11^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.983097162\)
\(L(\frac12)\) \(\approx\) \(6.983097162\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + p T^{2} )^{4} \)
11 \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \)
13 \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
good2 \( ( 1 - 3 T^{2} + p^{2} T^{4} )^{4} \)
5 \( ( 1 + p^{2} T^{4} )^{4} \)
7 \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \)
17 \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{4} \)
19 \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \)
23 \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{4} \)
31 \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \)
37 \( ( 1 - p T^{2} )^{8} \)
41 \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{4} \)
43 \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{4} \)
47 \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{4} \)
53 \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \)
59 \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{4} \)
61 \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \)
67 \( ( 1 - 104 T^{2} + p^{2} T^{4} )^{4} \)
71 \( ( 1 + 102 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{4} \)
79 \( ( 1 - 148 T^{2} + p^{2} T^{4} )^{4} \)
83 \( ( 1 - 162 T^{2} + p^{2} T^{4} )^{4} \)
89 \( ( 1 + 168 T^{2} + p^{2} T^{4} )^{4} \)
97 \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{4} \)
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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

−5.07554318920265461928165399321, −5.00479073195260296660613343641, −4.86470556934807182768937891741, −4.38725475015408138264331021898, −4.24838215781594132499593445922, −4.20713247798820858446140678073, −4.01044065715901894027728806960, −3.73418325082588303900545439535, −3.41727692964786015616321500787, −3.38943196111494385344942694132, −3.31793567612243685327676257964, −3.07818691102063799227753410754, −2.91334775514817237785793579861, −2.88939704245747309173082413708, −2.69385462098198913502075904767, −2.67657685307763313167054515563, −2.38675107297204637427583589064, −2.31235355529983806978621431062, −2.12957214607603629877565854009, −1.89472155857754178365895978694, −1.82709154149000353920501128067, −1.56043987756230250197157897733, −1.19583584977527851008393265982, −1.08088981180730255606093189827, −0.29784095610922973919854403482, 0.29784095610922973919854403482, 1.08088981180730255606093189827, 1.19583584977527851008393265982, 1.56043987756230250197157897733, 1.82709154149000353920501128067, 1.89472155857754178365895978694, 2.12957214607603629877565854009, 2.31235355529983806978621431062, 2.38675107297204637427583589064, 2.67657685307763313167054515563, 2.69385462098198913502075904767, 2.88939704245747309173082413708, 2.91334775514817237785793579861, 3.07818691102063799227753410754, 3.31793567612243685327676257964, 3.38943196111494385344942694132, 3.41727692964786015616321500787, 3.73418325082588303900545439535, 4.01044065715901894027728806960, 4.20713247798820858446140678073, 4.24838215781594132499593445922, 4.38725475015408138264331021898, 4.86470556934807182768937891741, 5.00479073195260296660613343641, 5.07554318920265461928165399321

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

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