Properties

Degree 16
Conductor $ 2^{40} \cdot 3^{24} \cdot 7^{8} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 20·25-s − 56·37-s − 4·49-s + 16·61-s + 64·73-s + 56·109-s − 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 80·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 4·25-s − 9.20·37-s − 4/7·49-s + 2.04·61-s + 7.49·73-s + 5.36·109-s − 2.90·121-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 − 6.15·169-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{40} \cdot 3^{24} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6048} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{40} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $0.02513588820$
$L(\frac12)$  $\approx$  $0.02513588820$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( ( 1 + T^{2} )^{4} \)
good5 \( ( 1 - 2 p T^{2} + 51 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 16 T^{2} + 210 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{4} \)
17 \( ( 1 - 58 T^{2} + 1395 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - 56 T^{2} + 1410 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 - 56 T^{2} + 1602 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 104 T^{2} + 4530 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 7 T + p T^{2} )^{8} \)
41 \( ( 1 - 122 T^{2} + 6867 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 46 T^{2} + 2283 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 118 T^{2} + 7515 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 172 T^{2} + 12630 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + 70 T^{2} + 5787 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 4 T + 72 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \)
67 \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \)
71 \( ( 1 + p T^{2} )^{8} \)
73 \( ( 1 - 16 T + 186 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \)
79 \( ( 1 - 62 T^{2} + 10539 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 182 T^{2} + 21195 T^{4} + 182 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{4} \)
97 \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.28532678419523672433887537579, −3.26167052563177931259275649550, −3.17686898089117780141469024172, −3.04338710699255371533246046680, −2.84277115674060600909801294506, −2.71783255657289928031714124386, −2.57881324194787748312886762594, −2.55513351075288154513406793566, −2.32560024535311574376018493353, −2.25641014701022329842338303078, −2.12685481278895341299278794509, −2.03017784336611726609759652918, −1.83603037776665765298946941370, −1.83312004451942824902122255991, −1.79573951910634519634507928737, −1.58037579211822471829780583338, −1.33666832799035833835007887596, −1.22124955038903024331015413269, −1.20554259665473607627727570593, −1.00396151521587073661790763150, −0.76277133495505136671894290280, −0.54380689866302118512548216040, −0.54378475954140263886431729626, −0.43651893917590036738013617165, −0.008677174784077116089984647120, 0.008677174784077116089984647120, 0.43651893917590036738013617165, 0.54378475954140263886431729626, 0.54380689866302118512548216040, 0.76277133495505136671894290280, 1.00396151521587073661790763150, 1.20554259665473607627727570593, 1.22124955038903024331015413269, 1.33666832799035833835007887596, 1.58037579211822471829780583338, 1.79573951910634519634507928737, 1.83312004451942824902122255991, 1.83603037776665765298946941370, 2.03017784336611726609759652918, 2.12685481278895341299278794509, 2.25641014701022329842338303078, 2.32560024535311574376018493353, 2.55513351075288154513406793566, 2.57881324194787748312886762594, 2.71783255657289928031714124386, 2.84277115674060600909801294506, 3.04338710699255371533246046680, 3.17686898089117780141469024172, 3.26167052563177931259275649550, 3.28532678419523672433887537579

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

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