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Properties

Label 108-1944e54-1.1-c0e54-0-0
Degree $108$
Conductor $3.887\times 10^{177}$
Sign $1$
Analytic cond. $0.195021$
Root an. cond. $0.984978$
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  + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 263-s + 269-s + ⋯
L(s)  = 1  + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 263-s + 269-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{162} \cdot 3^{270}\right)^{s/2} \, \Gamma_{\C}(s)^{54} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{162} \cdot 3^{270}\right)^{s/2} \, \Gamma_{\C}(s)^{54} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(108\)
Conductor: \(2^{162} \cdot 3^{270}\)
Sign: $1$
Analytic conductor: \(0.195021\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((108,\ 2^{162} \cdot 3^{270} ,\ ( \ : [0]^{54} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7447240606\)
\(L(\frac12)\) \(\approx\) \(0.7447240606\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{27} + T^{54} \)
3 \( 1 + T^{27} + T^{54} \)
good5 \( ( 1 - T^{27} + T^{54} )( 1 + T^{27} + T^{54} ) \)
7 \( ( 1 - T^{27} + T^{54} )( 1 + T^{27} + T^{54} ) \)
11 \( ( 1 + T^{9} + T^{18} )^{3}( 1 + T^{27} + T^{54} ) \)
13 \( ( 1 - T^{27} + T^{54} )( 1 + T^{27} + T^{54} ) \)
17 \( ( 1 + T^{27} + T^{54} )^{2} \)
19 \( ( 1 + T^{27} + T^{54} )^{2} \)
23 \( ( 1 - T^{27} + T^{54} )( 1 + T^{27} + T^{54} ) \)
29 \( ( 1 - T^{27} + T^{54} )( 1 + T^{27} + T^{54} ) \)
31 \( ( 1 - T^{27} + T^{54} )( 1 + T^{27} + T^{54} ) \)
37 \( ( 1 - T^{9} + T^{18} )^{3}( 1 + T^{9} + T^{18} )^{3} \)
41 \( ( 1 + T^{9} + T^{18} )^{3}( 1 + T^{27} + T^{54} ) \)
43 \( ( 1 + T^{9} + T^{18} )^{3}( 1 + T^{27} + T^{54} ) \)
47 \( ( 1 - T^{27} + T^{54} )( 1 + T^{27} + T^{54} ) \)
53 \( ( 1 - T^{3} + T^{6} )^{9}( 1 + T^{3} + T^{6} )^{9} \)
59 \( ( 1 + T^{3} + T^{6} )^{9}( 1 + T^{27} + T^{54} ) \)
61 \( ( 1 - T^{27} + T^{54} )( 1 + T^{27} + T^{54} ) \)
67 \( ( 1 + T^{9} + T^{18} )^{3}( 1 + T^{27} + T^{54} ) \)
71 \( ( 1 - T^{9} + T^{18} )^{3}( 1 + T^{9} + T^{18} )^{3} \)
73 \( ( 1 + T^{27} + T^{54} )^{2} \)
79 \( ( 1 - T^{27} + T^{54} )( 1 + T^{27} + T^{54} ) \)
83 \( ( 1 + T^{27} + T^{54} )^{2} \)
89 \( ( 1 + T^{3} + T^{6} )^{9}( 1 + T^{9} + T^{18} )^{3} \)
97 \( ( 1 + T^{9} + T^{18} )^{3}( 1 + T^{27} + T^{54} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{108} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−1.23610383306555588375937190546, −1.22744776147458492947223805672, −1.21830254246718388931659877743, −1.10685100496249315764644457687, −1.10079551765817248289829685509, −1.05147743800247709565359622250, −1.03392599502937352250913684613, −1.03259667397494363354977850911, −1.02313463253135801641108865917, −1.00026187540506656037902558276, −0.946047597237425085683582582645, −0.908018146621080043594201248193, −0.903829603765475101592759811790, −0.881552458203894150034951593019, −0.851709325262539438576015355766, −0.797787168087831561651414219755, −0.73577142906083152779594243623, −0.62626866193901133151067265249, −0.58260662461013936422238210551, −0.52222977977739880857009227520, −0.47536756092839370101794135007, −0.43664253170800265323186546969, −0.34696566902567202144394028470, −0.30313907620589986566798070849, −0.29214751300694434754645117651, 0.29214751300694434754645117651, 0.30313907620589986566798070849, 0.34696566902567202144394028470, 0.43664253170800265323186546969, 0.47536756092839370101794135007, 0.52222977977739880857009227520, 0.58260662461013936422238210551, 0.62626866193901133151067265249, 0.73577142906083152779594243623, 0.797787168087831561651414219755, 0.851709325262539438576015355766, 0.881552458203894150034951593019, 0.903829603765475101592759811790, 0.908018146621080043594201248193, 0.946047597237425085683582582645, 1.00026187540506656037902558276, 1.02313463253135801641108865917, 1.03259667397494363354977850911, 1.03392599502937352250913684613, 1.05147743800247709565359622250, 1.10079551765817248289829685509, 1.10685100496249315764644457687, 1.21830254246718388931659877743, 1.22744776147458492947223805672, 1.23610383306555588375937190546

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

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