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Properties

Label 2-72-24.5-c4-0-0
Degree 22
Conductor 7272
Sign −0.792+0.609i-0.792 + 0.609i
Analytic cond. 7.442637.44263
Root an. cond. 2.728112.72811
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.64 + 3.64i)2-s + (−10.5 + 11.9i)4-s − 21.8·5-s + 2.75·7-s + (−61.1 − 18.9i)8-s + (−35.9 − 79.6i)10-s − 173.·11-s − 121. i·13-s + (4.52 + 10.0i)14-s + (−31.5 − 254. i)16-s + 347. i·17-s + 88.8i·19-s + (231. − 261. i)20-s + (−285. − 633. i)22-s + 230. i·23-s + ⋯
L(s)  = 1  + (0.411 + 0.911i)2-s + (−0.662 + 0.749i)4-s − 0.873·5-s + 0.0561·7-s + (−0.955 − 0.295i)8-s + (−0.359 − 0.796i)10-s − 1.43·11-s − 0.721i·13-s + (0.0230 + 0.0511i)14-s + (−0.123 − 0.992i)16-s + 1.20i·17-s + 0.246i·19-s + (0.578 − 0.654i)20-s + (−0.590 − 1.30i)22-s + 0.436i·23-s + ⋯

Functional equation

Λ(s)=(72s/2 ΓC(s) L(s)=((−0.792+0.609i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 + 0.609i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(72s/2 ΓC(s+2) L(s)=((−0.792+0.609i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7272    =    23⋅322^{3} \cdot 3^{2}
Sign: −0.792+0.609i-0.792 + 0.609i
Analytic conductor: 7.442637.44263
Root analytic conductor: 2.728112.72811
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ72(53,⋅)\chi_{72} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 72, ( :2), −0.792+0.609i)(2,\ 72,\ (\ :2),\ -0.792 + 0.609i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.152714−0.449295i0.152714 - 0.449295i
L(12)L(\frac12) ≈\approx 0.152714−0.449295i0.152714 - 0.449295i
L(3)L(3) 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+(−1.64−3.64i)T 1 + (-1.64 - 3.64i)T
3 1 1
good5 1+21.8T+625T2 1 + 21.8T + 625T^{2}
7 1−2.75T+2.40e3T2 1 - 2.75T + 2.40e3T^{2}
11 1+173.T+1.46e4T2 1 + 173.T + 1.46e4T^{2}
13 1+121.iT−2.85e4T2 1 + 121. iT - 2.85e4T^{2}
17 1−347.iT−8.35e4T2 1 - 347. iT - 8.35e4T^{2}
19 1−88.8iT−1.30e5T2 1 - 88.8iT - 1.30e5T^{2}
23 1−230.iT−2.79e5T2 1 - 230. iT - 2.79e5T^{2}
29 1+1.31e3T+7.07e5T2 1 + 1.31e3T + 7.07e5T^{2}
31 1−1.67e3T+9.23e5T2 1 - 1.67e3T + 9.23e5T^{2}
37 1−1.40e3iT−1.87e6T2 1 - 1.40e3iT - 1.87e6T^{2}
41 1−2.07e3iT−2.82e6T2 1 - 2.07e3iT - 2.82e6T^{2}
43 1−3.31e3iT−3.41e6T2 1 - 3.31e3iT - 3.41e6T^{2}
47 1+1.47e3iT−4.87e6T2 1 + 1.47e3iT - 4.87e6T^{2}
53 1−579.T+7.89e6T2 1 - 579.T + 7.89e6T^{2}
59 1−2.27e3T+1.21e7T2 1 - 2.27e3T + 1.21e7T^{2}
61 1+5.79e3iT−1.38e7T2 1 + 5.79e3iT - 1.38e7T^{2}
67 1+2.80e3iT−2.01e7T2 1 + 2.80e3iT - 2.01e7T^{2}
71 1+7.67e3iT−2.54e7T2 1 + 7.67e3iT - 2.54e7T^{2}
73 1+8.36e3T+2.83e7T2 1 + 8.36e3T + 2.83e7T^{2}
79 1−1.01e3T+3.89e7T2 1 - 1.01e3T + 3.89e7T^{2}
83 1−305.T+4.74e7T2 1 - 305.T + 4.74e7T^{2}
89 1+1.94e3iT−6.27e7T2 1 + 1.94e3iT - 6.27e7T^{2}
97 1+7.67e3T+8.85e7T2 1 + 7.67e3T + 8.85e7T^{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

−14.93808131508743217273316911123, −13.39287906153429823430058486527, −12.71223711878559131121383554035, −11.42572349647126765241404324483, −9.977817001450070681593776674069, −8.153955789168717193349541100234, −7.80101673864122612253586573253, −6.11496841790381336208261214033, −4.78758776286810002850183004184, −3.30984354869934721500138517098, 0.20992042405445104794256612375, 2.54358396823476600768049226270, 4.12912012051496936497816895712, 5.39624670068343400606659951126, 7.35639891449711747626393444414, 8.764738546917228436149106294740, 10.09132411076119579420488709710, 11.21511274949741262109865217617, 11.99255497678330135316046588315, 13.13286207517023139438004258628

Graph of the ZZ-function along the critical line