Properties

Label 16-12e24-1.1-c1e8-0-0
Degree $16$
Conductor $7.950\times 10^{25}$
Sign $1$
Analytic cond. $1.31391\times 10^{9}$
Root an. cond. $3.71458$
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  + 10·25-s + 36·41-s − 2·49-s + 64·73-s + 96·89-s − 4·97-s + 12·113-s − 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 2·25-s + 5.62·41-s − 2/7·49-s + 7.49·73-s + 10.1·89-s − 0.406·97-s + 1.12·113-s − 2.36·121-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 − 1.69·169-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(1.31391\times 10^{9}\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4452529887\)
\(L(\frac12)\) \(\approx\) \(0.4452529887\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( ( 1 - p T^{2} )^{4}( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
7 \( ( 1 + T^{2} - 48 T^{4} + p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 + 11 T^{2} - 48 T^{4} + 11 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( ( 1 + p T^{2} )^{8} \)
19 \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \)
23 \( ( 1 - 31 T^{2} + 432 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + 43 T^{2} + 1008 T^{4} + 43 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 47 T^{2} + 1248 T^{4} - 47 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \)
41 \( ( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{4} \)
43 \( ( 1 + 37 T^{2} - 480 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 - 79 T^{2} + 4032 T^{4} - 79 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{4} \)
59 \( ( 1 + 109 T^{2} + 8400 T^{4} + 109 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 13 T^{2} - 3552 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( ( 1 - 109 T^{2} + p^{2} T^{4} )^{2}( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \)
71 \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 - 8 T + p T^{2} )^{8} \)
79 \( ( 1 - 143 T^{2} + 14208 T^{4} - 143 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 157 T^{2} + 17760 T^{4} + 157 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 12 T + p T^{2} )^{8} \)
97 \( ( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \)
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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

−3.91067262517487564503949347359, −3.70849048308428593812936501319, −3.62658144849637882857276457357, −3.50578539339254336526733859257, −3.45153655713605095835442867697, −3.36440804851288505089115091758, −3.19306582013311400009082413617, −3.08522870401622026305255721537, −3.05944668302515222101841018971, −2.79601580276702422997081623487, −2.39144527626817456223326261830, −2.38950210778554814943273164673, −2.25028152991399401013664697368, −2.23024440273012056275551189992, −2.21558523466826844170685379738, −2.20014901839750596189794580487, −1.87969480673275856951826264230, −1.72621938999425751733537117767, −1.20846267383728973074569124054, −1.08427281690167993489433236308, −0.931299665593158425479742861253, −0.857274569487811462988033166934, −0.800997812469302089562501330940, −0.78642157028277398877706087950, −0.04633864865943620649984461345, 0.04633864865943620649984461345, 0.78642157028277398877706087950, 0.800997812469302089562501330940, 0.857274569487811462988033166934, 0.931299665593158425479742861253, 1.08427281690167993489433236308, 1.20846267383728973074569124054, 1.72621938999425751733537117767, 1.87969480673275856951826264230, 2.20014901839750596189794580487, 2.21558523466826844170685379738, 2.23024440273012056275551189992, 2.25028152991399401013664697368, 2.38950210778554814943273164673, 2.39144527626817456223326261830, 2.79601580276702422997081623487, 3.05944668302515222101841018971, 3.08522870401622026305255721537, 3.19306582013311400009082413617, 3.36440804851288505089115091758, 3.45153655713605095835442867697, 3.50578539339254336526733859257, 3.62658144849637882857276457357, 3.70849048308428593812936501319, 3.91067262517487564503949347359

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

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