Properties

Degree $16$
Conductor $5.071\times 10^{24}$
Sign $1$
Motivic weight $0$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·11-s − 16-s + 8·71-s + 2·81-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  + 4·11-s − 16-s + 8·71-s + 2·81-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{16} \cdot 7^{16}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{1225} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{16} \cdot 7^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.439588194\)
\(L(\frac12)\) \(\approx\) \(1.439588194\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
3 \( ( 1 - T^{4} + T^{8} )^{2} \)
11 \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \)
13 \( ( 1 + T^{4} )^{4} \)
17 \( ( 1 - T^{4} + T^{8} )^{2} \)
19 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
23 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
29 \( ( 1 - T^{2} + T^{4} )^{4} \)
31 \( ( 1 - T^{2} + T^{4} )^{4} \)
37 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
41 \( ( 1 + T^{2} )^{8} \)
43 \( ( 1 - T^{4} + T^{8} )^{2} \)
47 \( ( 1 - T^{4} + T^{8} )^{2} \)
53 \( ( 1 - T^{4} + T^{8} )^{2} \)
59 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
61 \( ( 1 - T^{2} + T^{4} )^{4} \)
67 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
71 \( ( 1 - T + T^{2} )^{8} \)
73 \( ( 1 - T^{4} + T^{8} )^{2} \)
79 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
83 \( ( 1 + T^{4} )^{4} \)
89 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
97 \( ( 1 + T^{4} )^{4} \)
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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

−4.36941610548638409971685415239, −4.35851928887375527034853727680, −3.96647107914123006314442429121, −3.96349469404813819988081851981, −3.86968123972594648253967970894, −3.84273784607651024876426960417, −3.71606341840407907275218434847, −3.41650230896378843816209700367, −3.31354517513423685009325684271, −3.29867726749598220788928904031, −3.28801632691664035382696438772, −3.06351547788081039922688455579, −2.89298250954082879827283951574, −2.46475113049638324653093542350, −2.31278192804378191283822101731, −2.22007634335971687663320888564, −2.17338782806571675765020096539, −2.11991578274121113102865490538, −2.02665743597294185186957471526, −1.80517201745387408000729403296, −1.27619977569278877705028896279, −1.19578509012443993254680863876, −1.14208435777253230841162635923, −1.06208500442160328966174418076, −0.795073279066333925922302132925, 0.795073279066333925922302132925, 1.06208500442160328966174418076, 1.14208435777253230841162635923, 1.19578509012443993254680863876, 1.27619977569278877705028896279, 1.80517201745387408000729403296, 2.02665743597294185186957471526, 2.11991578274121113102865490538, 2.17338782806571675765020096539, 2.22007634335971687663320888564, 2.31278192804378191283822101731, 2.46475113049638324653093542350, 2.89298250954082879827283951574, 3.06351547788081039922688455579, 3.28801632691664035382696438772, 3.29867726749598220788928904031, 3.31354517513423685009325684271, 3.41650230896378843816209700367, 3.71606341840407907275218434847, 3.84273784607651024876426960417, 3.86968123972594648253967970894, 3.96349469404813819988081851981, 3.96647107914123006314442429121, 4.35851928887375527034853727680, 4.36941610548638409971685415239

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

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