Properties

Label 16-1575e8-1.1-c1e8-0-11
Degree $16$
Conductor $3.787\times 10^{25}$
Sign $1$
Analytic cond. $6.25837\times 10^{8}$
Root an. cond. $3.54632$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·16-s − 28·49-s − 64·79-s + ⋯
L(s)  = 1  − 1/2·16-s − 4·49-s − 7.20·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 5^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(6.25837\times 10^{8}\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.022498334\)
\(L(\frac12)\) \(\approx\) \(3.022498334\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( ( 1 + p T^{2} )^{4} \)
good2 \( ( 1 + T^{4} + p^{4} T^{8} )^{2} \)
11 \( ( 1 - 206 T^{4} + p^{4} T^{8} )^{2} \)
13 \( ( 1 + p T^{2} )^{8} \)
17 \( ( 1 - p T^{2} )^{8} \)
19 \( ( 1 - p T^{2} )^{8} \)
23 \( ( 1 - 734 T^{4} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + 1234 T^{4} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - p T^{2} )^{8} \)
37 \( ( 1 - 6 T + p T^{2} )^{4}( 1 + 6 T + p T^{2} )^{4} \)
41 \( ( 1 + p T^{2} )^{8} \)
43 \( ( 1 - 12 T + p T^{2} )^{4}( 1 + 12 T + p T^{2} )^{4} \)
47 \( ( 1 - p T^{2} )^{8} \)
53 \( ( 1 - 5582 T^{4} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + p T^{2} )^{8} \)
61 \( ( 1 - p T^{2} )^{8} \)
67 \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{4} \)
71 \( ( 1 + 2914 T^{4} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + p T^{2} )^{8} \)
79 \( ( 1 + 8 T + p T^{2} )^{8} \)
83 \( ( 1 - p T^{2} )^{8} \)
89 \( ( 1 + p T^{2} )^{8} \)
97 \( ( 1 + p T^{2} )^{8} \)
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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

−3.95969284940731362510645684462, −3.95315213779608048476792691869, −3.94752910368177138150104479064, −3.49854734237569570013278648612, −3.45740609466890724981006101457, −3.34663427679958673929933228251, −3.30030712967621958064023777741, −3.02936952948878981933851872273, −2.91597936321734709369263731858, −2.79394642864790419460320192274, −2.78141131727146699865099352614, −2.55390539673096436576432095695, −2.45256342719733179743077281658, −2.36847247202374454391357951257, −2.07389253187590712904146851414, −1.90262629014249205695760273683, −1.77743632075687172682451789767, −1.53049402761679648135677625605, −1.48699860281010991571154022895, −1.30355256068089636526925304285, −1.28240743147851084461272159895, −1.05985233720988395702603278197, −0.40720742392417692765694078767, −0.35393755709541368543259235034, −0.32598910227156630652229399578, 0.32598910227156630652229399578, 0.35393755709541368543259235034, 0.40720742392417692765694078767, 1.05985233720988395702603278197, 1.28240743147851084461272159895, 1.30355256068089636526925304285, 1.48699860281010991571154022895, 1.53049402761679648135677625605, 1.77743632075687172682451789767, 1.90262629014249205695760273683, 2.07389253187590712904146851414, 2.36847247202374454391357951257, 2.45256342719733179743077281658, 2.55390539673096436576432095695, 2.78141131727146699865099352614, 2.79394642864790419460320192274, 2.91597936321734709369263731858, 3.02936952948878981933851872273, 3.30030712967621958064023777741, 3.34663427679958673929933228251, 3.45740609466890724981006101457, 3.49854734237569570013278648612, 3.94752910368177138150104479064, 3.95315213779608048476792691869, 3.95969284940731362510645684462

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

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