Properties

Degree 16
Conductor $ 2^{48} \cdot 3^{16} \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  + 24·25-s − 16·29-s + 16·37-s + 4·49-s − 16·53-s − 48·109-s − 48·113-s + 72·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 24·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 24/5·25-s − 2.97·29-s + 2.63·37-s + 4/7·49-s − 2.19·53-s − 4.59·109-s − 4.51·113-s + 6.54·121-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 + 1.84·169-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{48} \cdot 3^{16} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4032} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{48} \cdot 3^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $4.069586291$
$L(\frac12)$  $\approx$  $4.069586291$
$L(\frac{3}{2})$   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 \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
good5 \( ( 1 - 12 T^{2} + 78 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \)
13 \( ( 1 - 12 T^{2} + 366 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
19 \( ( 1 + 36 T^{2} + 1038 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 - 20 T^{2} + 646 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \)
31 \( ( 1 + 60 T^{2} + 2310 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \)
41 \( ( 1 - 4 T^{2} + 3238 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 100 T^{2} + 5686 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 124 T^{2} + 7750 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 + 2 T + p T^{2} )^{8} \)
59 \( ( 1 + 132 T^{2} + 10926 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 236 T^{2} + 21358 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( ( 1 - 4 T^{2} - 3818 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - 196 T^{2} + 18534 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 132 T^{2} + 8742 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
79 \( ( 1 - 100 T^{2} + 11782 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 196 T^{2} + 19150 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 324 T^{2} + 41958 T^{4} - 324 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 68 T^{2} + 19462 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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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

−3.40265303953789240958207852430, −3.37597070907559862748649076759, −3.21977941674041807098400259881, −3.11056167959962020054233720459, −2.90366040021137227855991874528, −2.89690700382986480799362859085, −2.82703216514415358211034722334, −2.67290882222862017130548317206, −2.49093189840015606485300643980, −2.44024087931290368483116396851, −2.33183834053790796068972164540, −2.32487986298231511032584345175, −2.24793593314390148681769576845, −1.73413510574749569023784681742, −1.58136203424289779083866351139, −1.58107184043815806772597538864, −1.57369232916943217919340366304, −1.46789073646230597870956775390, −1.15085227969323348129966864271, −1.14233314700413731101887736021, −0.959768828220903411829932930217, −0.61508377128322386821236700062, −0.53312484116502612299258074107, −0.45146470965578731625101749957, −0.13124100875980126127178840484, 0.13124100875980126127178840484, 0.45146470965578731625101749957, 0.53312484116502612299258074107, 0.61508377128322386821236700062, 0.959768828220903411829932930217, 1.14233314700413731101887736021, 1.15085227969323348129966864271, 1.46789073646230597870956775390, 1.57369232916943217919340366304, 1.58107184043815806772597538864, 1.58136203424289779083866351139, 1.73413510574749569023784681742, 2.24793593314390148681769576845, 2.32487986298231511032584345175, 2.33183834053790796068972164540, 2.44024087931290368483116396851, 2.49093189840015606485300643980, 2.67290882222862017130548317206, 2.82703216514415358211034722334, 2.89690700382986480799362859085, 2.90366040021137227855991874528, 3.11056167959962020054233720459, 3.21977941674041807098400259881, 3.37597070907559862748649076759, 3.40265303953789240958207852430

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

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