Properties

Label 16-4032e8-1.1-c1e8-0-1
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  − 32·25-s + 20·49-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 + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯
L(s)  = 1  − 6.39·25-s + 20/7·49-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 + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-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

\(L(1)\) \(\approx\) \(0.3211327097\)
\(L(\frac12)\) \(\approx\) \(0.3211327097\)
\(L(\frac{3}{2})\) 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 \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
good5 \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \)
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

−3.56498202363410212107600969051, −3.34430355010386565986538541540, −3.30951505729650825148447340301, −3.22639201520828456143403045164, −2.98295262326766249944305955094, −2.91205459944702433095297288580, −2.76610625983105064212462993682, −2.58576901583342993414847325921, −2.40581754919085392437872356724, −2.39705750281945139763259569666, −2.34134910228465659913863843605, −2.21142405776857027013473220163, −1.98307531043502486974613685021, −1.87428668967976392852758189548, −1.77977472075599113627954142308, −1.77393592739132982523766256422, −1.64925633260620435728135738152, −1.37485016163143301901602881663, −1.20376026882145597606574023517, −1.12603938735552139829015927113, −0.75514871342674771733566827741, −0.71987696797490580908168032134, −0.47541861450766493886441763221, −0.35494873524933593420990089930, −0.05000685128759226873769255427, 0.05000685128759226873769255427, 0.35494873524933593420990089930, 0.47541861450766493886441763221, 0.71987696797490580908168032134, 0.75514871342674771733566827741, 1.12603938735552139829015927113, 1.20376026882145597606574023517, 1.37485016163143301901602881663, 1.64925633260620435728135738152, 1.77393592739132982523766256422, 1.77977472075599113627954142308, 1.87428668967976392852758189548, 1.98307531043502486974613685021, 2.21142405776857027013473220163, 2.34134910228465659913863843605, 2.39705750281945139763259569666, 2.40581754919085392437872356724, 2.58576901583342993414847325921, 2.76610625983105064212462993682, 2.91205459944702433095297288580, 2.98295262326766249944305955094, 3.22639201520828456143403045164, 3.30951505729650825148447340301, 3.34430355010386565986538541540, 3.56498202363410212107600969051

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

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