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Properties

Label 128-3459e64-1.1-c0e64-0-0
Degree $128$
Conductor $3.111\times 10^{226}$
Sign $1$
Analytic cond. $1.49586\times 10^{15}$
Root an. cond. $1.31387$
Motivic weight $0$
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  − 64·67-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 16·169-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 + 251-s + 257-s + ⋯
L(s)  = 1  − 64·67-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 16·169-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 + 251-s + 257-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{64} \cdot 1153^{64}\right)^{s/2} \, \Gamma_{\C}(s)^{64} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{64} \cdot 1153^{64}\right)^{s/2} \, \Gamma_{\C}(s)^{64} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(128\)
Conductor: \(3^{64} \cdot 1153^{64}\)
Sign: $1$
Analytic conductor: \(1.49586\times 10^{15}\)
Root analytic conductor: \(1.31387\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((128,\ 3^{64} \cdot 1153^{64} ,\ ( \ : [0]^{64} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04570354341\)
\(L(\frac12)\) \(\approx\) \(0.04570354341\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T^{32} + T^{64} \)
1153 \( ( 1 + T^{32} )^{2} \)
good2 \( ( 1 - T^{16} + T^{32} )^{4} \)
5 \( 1 - T^{64} + T^{128} \)
7 \( ( 1 - T^{8} + T^{16} )^{4}( 1 - T^{32} + T^{64} ) \)
11 \( ( 1 - T^{32} + T^{64} )^{2} \)
13 \( ( 1 - T^{2} + T^{4} )^{16}( 1 - T^{16} + T^{32} )^{2} \)
17 \( 1 - T^{64} + T^{128} \)
19 \( ( 1 + T^{8} )^{8}( 1 - T^{32} + T^{64} ) \)
23 \( ( 1 - T^{4} + T^{8} )^{16} \)
29 \( ( 1 - T^{32} + T^{64} )^{2} \)
31 \( ( 1 + T^{16} )^{4}( 1 - T^{32} + T^{64} ) \)
37 \( ( 1 + T^{16} )^{4}( 1 + T^{32} )^{2} \)
41 \( ( 1 - T^{32} + T^{64} )^{2} \)
43 \( ( 1 + T^{32} )^{4} \)
47 \( ( 1 - T^{2} + T^{4} )^{32} \)
53 \( ( 1 + T^{2} )^{64} \)
59 \( ( 1 + T^{64} )^{2} \)
61 \( ( 1 - T^{16} + T^{32} )^{2}( 1 - T^{32} + T^{64} ) \)
67 \( ( 1 + T )^{64}( 1 + T^{4} )^{16} \)
71 \( ( 1 + T^{64} )^{2} \)
73 \( ( 1 - T^{16} + T^{32} )^{2}( 1 + T^{32} )^{2} \)
79 \( ( 1 - T^{16} + T^{32} )^{2}( 1 - T^{32} + T^{64} ) \)
83 \( 1 - T^{64} + T^{128} \)
89 \( ( 1 - T^{32} + T^{64} )^{2} \)
97 \( ( 1 + T^{32} )^{2}( 1 - T^{32} + T^{64} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{128} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−1.01593606308255673709229054267, −0.997320863694616792935161297097, −0.965301229340500656489817038222, −0.957125624251863181439641999084, −0.956599991490633780781757029018, −0.913134151275009547700492830767, −0.882403712295727968115694734057, −0.833684601488239696507731460976, −0.818501375253735708465233249289, −0.803068834716076944904485256244, −0.76602906159214536930015766312, −0.63782663163373403810704703711, −0.54574897662775947953528507903, −0.52001149859383968926000260793, −0.49659398135411841643493519040, −0.47874048366563199556253694028, −0.46552672939612106099252322820, −0.42830075356643388313664448971, −0.34623155658898744027581723752, −0.32537034068287857256246594901, −0.25531913673854252419803644478, −0.19669255327807860767185728779, −0.18354460161192878671344806323, −0.16061595725742137190373787370, −0.12687266315040936661864870413, 0.12687266315040936661864870413, 0.16061595725742137190373787370, 0.18354460161192878671344806323, 0.19669255327807860767185728779, 0.25531913673854252419803644478, 0.32537034068287857256246594901, 0.34623155658898744027581723752, 0.42830075356643388313664448971, 0.46552672939612106099252322820, 0.47874048366563199556253694028, 0.49659398135411841643493519040, 0.52001149859383968926000260793, 0.54574897662775947953528507903, 0.63782663163373403810704703711, 0.76602906159214536930015766312, 0.803068834716076944904485256244, 0.818501375253735708465233249289, 0.833684601488239696507731460976, 0.882403712295727968115694734057, 0.913134151275009547700492830767, 0.956599991490633780781757029018, 0.957125624251863181439641999084, 0.965301229340500656489817038222, 0.997320863694616792935161297097, 1.01593606308255673709229054267

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

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