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Properties

Label 72-2627e36-1.1-c0e36-0-1
Degree $72$
Conductor $1.261\times 10^{123}$
Sign $1$
Analytic cond. $17148.2$
Root an. cond. $1.14500$
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  + 6·3-s + 3·5-s + 15·9-s + 18·15-s + 3·19-s + 6·25-s + 19·27-s − 9·29-s + 45·45-s + 18·57-s + 64-s + 36·75-s + 6·79-s + 9·81-s − 54·87-s + 3·89-s + 9·95-s + 6·109-s − 18·121-s + 6·125-s + 127-s + 131-s + 57·135-s + 137-s + 139-s − 27·145-s + 149-s + ⋯
L(s)  = 1  + 6·3-s + 3·5-s + 15·9-s + 18·15-s + 3·19-s + 6·25-s + 19·27-s − 9·29-s + 45·45-s + 18·57-s + 64-s + 36·75-s + 6·79-s + 9·81-s − 54·87-s + 3·89-s + 9·95-s + 6·109-s − 18·121-s + 6·125-s + 127-s + 131-s + 57·135-s + 137-s + 139-s − 27·145-s + 149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(37^{36} \cdot 71^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{36} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(37^{36} \cdot 71^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{36} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(72\)
Conductor: \(37^{36} \cdot 71^{36}\)
Sign: $1$
Analytic conductor: \(17148.2\)
Root analytic conductor: \(1.14500\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((72,\ 37^{36} \cdot 71^{36} ,\ ( \ : [0]^{36} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(56.02875326\)
\(L(\frac12)\) \(\approx\) \(56.02875326\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + T^{3} - T^{9} - T^{12} + T^{18} - T^{24} - T^{27} + T^{33} + T^{36} \)
71 \( ( 1 - T^{3} + T^{6} )^{6} \)
good2 \( ( 1 - T^{3} + T^{9} - T^{12} + T^{18} - T^{24} + T^{27} - T^{33} + T^{36} )( 1 + T^{3} - T^{9} - T^{12} + T^{18} - T^{24} - T^{27} + T^{33} + T^{36} ) \)
3 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{6}( 1 + T^{3} - T^{9} - T^{12} + T^{18} - T^{24} - T^{27} + T^{33} + T^{36} ) \)
5 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{3}( 1 + T^{3} - T^{9} - T^{12} + T^{18} - T^{24} - T^{27} + T^{33} + T^{36} ) \)
7 \( ( 1 - T^{3} + T^{6} )^{6}( 1 + T^{3} + T^{6} )^{6} \)
11 \( ( 1 - T + T^{2} )^{18}( 1 + T + T^{2} )^{18} \)
13 \( ( 1 - T^{6} + T^{12} )^{6} \)
17 \( ( 1 - T^{6} + T^{12} )^{6} \)
19 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{3}( 1 + T^{3} - T^{9} - T^{12} + T^{18} - T^{24} - T^{27} + T^{33} + T^{36} ) \)
23 \( ( 1 - T^{2} + T^{4} )^{18} \)
29 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{6}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{3} \)
31 \( ( 1 + T^{2} )^{36} \)
41 \( ( 1 - T^{3} + T^{6} )^{6}( 1 + T^{3} + T^{6} )^{6} \)
43 \( ( 1 - T^{3} + T^{9} - T^{12} + T^{18} - T^{24} + T^{27} - T^{33} + T^{36} )( 1 + T^{3} - T^{9} - T^{12} + T^{18} - T^{24} - T^{27} + T^{33} + T^{36} ) \)
47 \( ( 1 - T + T^{2} )^{18}( 1 + T + T^{2} )^{18} \)
53 \( ( 1 - T^{3} + T^{6} )^{6}( 1 + T^{3} + T^{6} )^{6} \)
59 \( ( 1 - T^{6} + T^{12} )^{6} \)
61 \( ( 1 - T^{6} + T^{12} )^{6} \)
67 \( ( 1 - T^{3} + T^{6} )^{6}( 1 + T^{3} + T^{6} )^{6} \)
73 \( ( 1 - T^{3} + T^{9} - T^{12} + T^{18} - T^{24} + T^{27} - T^{33} + T^{36} )^{2} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{6}( 1 - T^{3} + T^{9} - T^{12} + T^{18} - T^{24} + T^{27} - T^{33} + T^{36} ) \)
83 \( ( 1 - T^{3} + T^{9} - T^{12} + T^{18} - T^{24} + T^{27} - T^{33} + T^{36} )^{2} \)
89 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{3}( 1 + T^{3} - T^{9} - T^{12} + T^{18} - T^{24} - T^{27} + T^{33} + T^{36} ) \)
97 \( ( 1 - T^{2} + T^{4} )^{18} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{72} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−1.49859226909477908470772597005, −1.48055585565769007562170667810, −1.47069796034150324511150380682, −1.43049040155411036595729843544, −1.41676604297449765165407075416, −1.29545906740115457727722720512, −1.26340663878183122499238588721, −1.18304102447373703419573995979, −1.13234391799002613600011120245, −1.11953898147509331080033334187, −1.09850403921367990330981497680, −1.08918148716677933890791229900, −1.06090757569435863808399328254, −1.03609817794350902670699797878, −0.996741628074675479664759834384, −0.983136941076951444724600820598, −0.849175243010357357518409731273, −0.77296891596763072268577525205, −0.75963630400467246821375126202, −0.71359514189666058380556680891, −0.51212766527789766399162213667, −0.47265003119383053061610845045, −0.42091551378689514799245788617, −0.33554194654528656368351522092, −0.32984035093974302414382598683, 0.32984035093974302414382598683, 0.33554194654528656368351522092, 0.42091551378689514799245788617, 0.47265003119383053061610845045, 0.51212766527789766399162213667, 0.71359514189666058380556680891, 0.75963630400467246821375126202, 0.77296891596763072268577525205, 0.849175243010357357518409731273, 0.983136941076951444724600820598, 0.996741628074675479664759834384, 1.03609817794350902670699797878, 1.06090757569435863808399328254, 1.08918148716677933890791229900, 1.09850403921367990330981497680, 1.11953898147509331080033334187, 1.13234391799002613600011120245, 1.18304102447373703419573995979, 1.26340663878183122499238588721, 1.29545906740115457727722720512, 1.41676604297449765165407075416, 1.43049040155411036595729843544, 1.47069796034150324511150380682, 1.48055585565769007562170667810, 1.49859226909477908470772597005

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

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