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Properties

Label 2-72-8.3-c20-0-57
Degree 22
Conductor 7272
Sign 11
Analytic cond. 182.529182.529
Root an. cond. 13.510313.5103
Motivic weight 2020
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.02e3·2-s + 1.04e6·4-s − 1.07e9·8-s + 4.23e10·11-s + 1.09e12·16-s + 3.35e12·17-s − 1.01e12·19-s − 4.34e13·22-s + 9.53e13·25-s − 1.12e15·32-s − 3.43e15·34-s + 1.03e15·38-s + 2.54e16·41-s + 2.78e15·43-s + 4.44e16·44-s + 7.97e16·49-s − 9.76e16·50-s + 1.73e17·59-s + 1.15e18·64-s − 3.56e17·67-s + 3.51e18·68-s − 6.01e18·73-s − 1.06e18·76-s − 2.60e19·82-s + 3.10e19·83-s − 2.84e18·86-s − 4.55e19·88-s + ⋯
L(s)  = 1  − 2-s + 4-s − 8-s + 1.63·11-s + 16-s + 1.66·17-s − 0.165·19-s − 1.63·22-s + 25-s − 32-s − 1.66·34-s + 0.165·38-s + 1.89·41-s + 0.128·43-s + 1.63·44-s + 49-s − 50-s + 0.340·59-s + 64-s − 0.195·67-s + 1.66·68-s − 1.40·73-s − 0.165·76-s − 1.89·82-s + 1.99·83-s − 0.128·86-s − 1.63·88-s + ⋯

Functional equation

Λ(s)=(72s/2 ΓC(s) L(s)=( Λ(21−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(21-s) \end{aligned}
Λ(s)=(72s/2 ΓC(s+10) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7272    =    23⋅322^{3} \cdot 3^{2}
Sign: 11
Analytic conductor: 182.529182.529
Root analytic conductor: 13.510313.5103
Motivic weight: 2020
Rational: yes
Arithmetic: yes
Character: χ72(19,⋅)\chi_{72} (19, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 72, ( :10), 1)(2,\ 72,\ (\ :10),\ 1)

Particular Values

L(212)L(\frac{21}{2}) ≈\approx 1.9677581041.967758104
L(12)L(\frac12) ≈\approx 1.9677581041.967758104
L(11)L(11) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+p10T 1 + p^{10} T
3 1 1
good5 (1−p10T)(1+p10T) ( 1 - p^{10} T )( 1 + p^{10} T )
7 (1−p10T)(1+p10T) ( 1 - p^{10} T )( 1 + p^{10} T )
11 1−42383023726T+p20T2 1 - 42383023726 T + p^{20} T^{2}
13 (1−p10T)(1+p10T) ( 1 - p^{10} T )( 1 + p^{10} T )
17 1−3353535763774T+p20T2 1 - 3353535763774 T + p^{20} T^{2}
19 1+1014654432526T+p20T2 1 + 1014654432526 T + p^{20} T^{2}
23 (1−p10T)(1+p10T) ( 1 - p^{10} T )( 1 + p^{10} T )
29 (1−p10T)(1+p10T) ( 1 - p^{10} T )( 1 + p^{10} T )
31 (1−p10T)(1+p10T) ( 1 - p^{10} T )( 1 + p^{10} T )
37 (1−p10T)(1+p10T) ( 1 - p^{10} T )( 1 + p^{10} T )
41 1−25418071370591326T+p20T2 1 - 25418071370591326 T + p^{20} T^{2}
43 1−2781113986388498T+p20T2 1 - 2781113986388498 T + p^{20} T^{2}
47 (1−p10T)(1+p10T) ( 1 - p^{10} T )( 1 + p^{10} T )
53 (1−p10T)(1+p10T) ( 1 - p^{10} T )( 1 + p^{10} T )
59 1−173912197184497198T+p20T2 1 - 173912197184497198 T + p^{20} T^{2}
61 (1−p10T)(1+p10T) ( 1 - p^{10} T )( 1 + p^{10} T )
67 1+356137514166464974T+p20T2 1 + 356137514166464974 T + p^{20} T^{2}
71 (1−p10T)(1+p10T) ( 1 - p^{10} T )( 1 + p^{10} T )
73 1+6016717170316692574T+p20T2 1 + 6016717170316692574 T + p^{20} T^{2}
79 (1−p10T)(1+p10T) ( 1 - p^{10} T )( 1 + p^{10} T )
83 1−31022856480301602574T+p20T2 1 - 31022856480301602574 T + p^{20} T^{2}
89 1+61202446863210984674T+p20T2 1 + 61202446863210984674 T + p^{20} T^{2}
97 1+50009130514058267902T+p20T2 1 + 50009130514058267902 T + p^{20} T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.69101734496217098666848402140, −9.617155241152997180504483296189, −8.845081649352637385181319401699, −7.68334532172466686784337896805, −6.69234139085407194074509950030, −5.67647835646137525879441359004, −3.95579651647993289169161350965, −2.81385343414303355035035655005, −1.42926763420696392447469492068, −0.78378243989395013061315096680, 0.78378243989395013061315096680, 1.42926763420696392447469492068, 2.81385343414303355035035655005, 3.95579651647993289169161350965, 5.67647835646137525879441359004, 6.69234139085407194074509950030, 7.68334532172466686784337896805, 8.845081649352637385181319401699, 9.617155241152997180504483296189, 10.69101734496217098666848402140

Graph of the ZZ-function along the critical line