Properties

Label 16-448e8-1.1-c1e8-0-1
Degree $16$
Conductor $1.623\times 10^{21}$
Sign $1$
Analytic cond. $26818.9$
Root an. cond. $1.89137$
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  + 16·9-s − 32·25-s + 20·49-s + 124·81-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 512·225-s + 227-s + 229-s + ⋯
L(s)  = 1  + 16/3·9-s − 6.39·25-s + 20/7·49-s + 13.7·81-s − 3.63·121-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 + 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 − 34.1·225-s + 0.0663·227-s + 0.0660·229-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(26818.9\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.625734342\)
\(L(\frac12)\) \(\approx\) \(1.625734342\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
good3 \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{4} \)
5 \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \)
11 \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \)
13 \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \)
17 \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \)
19 \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \)
23 \( ( 1 - p T^{2} )^{8} \)
29 \( ( 1 - p T^{2} )^{8} \)
31 \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \)
37 \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \)
41 \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \)
43 \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{4} \)
47 \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \)
53 \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{4} \)
59 \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{4} \)
61 \( ( 1 + 104 T^{2} + p^{2} T^{4} )^{4} \)
67 \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \)
71 \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{4} \)
79 \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \)
83 \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{4} \)
89 \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \)
97 \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{4} \)
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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

−4.79088327549167270224957356193, −4.76638366109868051267427951782, −4.54255745672273668848283164285, −4.38639150391389180712307344272, −4.21610532038637666154304484555, −4.15990911221103853884477304436, −3.98826418511634455127760998951, −3.95771310850442420163057043496, −3.71223921972168417243472232252, −3.70181416077068708550471718865, −3.68078553753299007651912355844, −3.54429975884060915681907309625, −3.26212953349621916363413334122, −2.60907780948850379342388754623, −2.60741658325559565487405188492, −2.45271261588049143874118745241, −2.44566525331869703619509702309, −2.19703364262668787585036280148, −1.80743338757989986319870438555, −1.62992531547686236985356964255, −1.60231692087868306763915758852, −1.42943818177249937183192426260, −1.19565744091903906847901562434, −0.967315265489517001562289388018, −0.19655198445435576392837210199, 0.19655198445435576392837210199, 0.967315265489517001562289388018, 1.19565744091903906847901562434, 1.42943818177249937183192426260, 1.60231692087868306763915758852, 1.62992531547686236985356964255, 1.80743338757989986319870438555, 2.19703364262668787585036280148, 2.44566525331869703619509702309, 2.45271261588049143874118745241, 2.60741658325559565487405188492, 2.60907780948850379342388754623, 3.26212953349621916363413334122, 3.54429975884060915681907309625, 3.68078553753299007651912355844, 3.70181416077068708550471718865, 3.71223921972168417243472232252, 3.95771310850442420163057043496, 3.98826418511634455127760998951, 4.15990911221103853884477304436, 4.21610532038637666154304484555, 4.38639150391389180712307344272, 4.54255745672273668848283164285, 4.76638366109868051267427951782, 4.79088327549167270224957356193

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

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