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Properties

Label 2-72-24.11-c21-0-0
Degree 22
Conductor 7272
Sign −0.384−0.923i-0.384 - 0.923i
Analytic cond. 201.223201.223
Root an. cond. 14.185314.1853
Motivic weight 2121
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36e3 + 478. i)2-s + (1.63e6 − 1.30e6i)4-s − 1.38e7·5-s − 1.09e9i·7-s + (−1.61e9 + 2.57e9i)8-s + (1.88e10 − 6.60e9i)10-s − 4.42e10i·11-s + 9.88e10i·13-s + (5.22e11 + 1.49e12i)14-s + (9.78e11 − 4.28e12i)16-s + 4.30e12i·17-s − 6.19e12·19-s + (−2.26e13 + 1.80e13i)20-s + (2.11e13 + 6.05e13i)22-s + 1.44e14·23-s + ⋯
L(s)  = 1  + (−0.943 + 0.330i)2-s + (0.781 − 0.623i)4-s − 0.632·5-s − 1.46i·7-s + (−0.532 + 0.846i)8-s + (0.597 − 0.208i)10-s − 0.514i·11-s + 0.198i·13-s + (0.482 + 1.37i)14-s + (0.222 − 0.974i)16-s + 0.518i·17-s − 0.231·19-s + (−0.494 + 0.394i)20-s + (0.169 + 0.485i)22-s + 0.727·23-s + ⋯

Functional equation

Λ(s)=(72s/2 ΓC(s) L(s)=((−0.384−0.923i) Λ‾(22−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.384 - 0.923i)\, \overline{\Lambda}(22-s) \end{aligned}
Λ(s)=(72s/2 ΓC(s+21/2) L(s)=((−0.384−0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7272    =    23⋅322^{3} \cdot 3^{2}
Sign: −0.384−0.923i-0.384 - 0.923i
Analytic conductor: 201.223201.223
Root analytic conductor: 14.185314.1853
Motivic weight: 2121
Rational: no
Arithmetic: yes
Character: χ72(35,⋅)\chi_{72} (35, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 72, ( :21/2), −0.384−0.923i)(2,\ 72,\ (\ :21/2),\ -0.384 - 0.923i)

Particular Values

L(11)L(11) ≈\approx 0.020564067910.02056406791
L(12)L(\frac12) ≈\approx 0.020564067910.02056406791
L(232)L(\frac{23}{2}) 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+(1.36e3−478.i)T 1 + (1.36e3 - 478. i)T
3 1 1
good5 1+1.38e7T+4.76e14T2 1 + 1.38e7T + 4.76e14T^{2}
7 1+1.09e9iT−5.58e17T2 1 + 1.09e9iT - 5.58e17T^{2}
11 1+4.42e10iT−7.40e21T2 1 + 4.42e10iT - 7.40e21T^{2}
13 1−9.88e10iT−2.47e23T2 1 - 9.88e10iT - 2.47e23T^{2}
17 1−4.30e12iT−6.90e25T2 1 - 4.30e12iT - 6.90e25T^{2}
19 1+6.19e12T+7.14e26T2 1 + 6.19e12T + 7.14e26T^{2}
23 1−1.44e14T+3.94e28T2 1 - 1.44e14T + 3.94e28T^{2}
29 1+1.75e14T+5.13e30T2 1 + 1.75e14T + 5.13e30T^{2}
31 1+3.72e15iT−2.08e31T2 1 + 3.72e15iT - 2.08e31T^{2}
37 1+3.46e16iT−8.55e32T2 1 + 3.46e16iT - 8.55e32T^{2}
41 1+1.63e16iT−7.38e33T2 1 + 1.63e16iT - 7.38e33T^{2}
43 1+1.19e17T+2.00e34T2 1 + 1.19e17T + 2.00e34T^{2}
47 1+2.37e17T+1.30e35T2 1 + 2.37e17T + 1.30e35T^{2}
53 1−3.01e17T+1.62e36T2 1 - 3.01e17T + 1.62e36T^{2}
59 1+1.83e18iT−1.54e37T2 1 + 1.83e18iT - 1.54e37T^{2}
61 1−1.78e18iT−3.10e37T2 1 - 1.78e18iT - 3.10e37T^{2}
67 1+1.98e19T+2.22e38T2 1 + 1.98e19T + 2.22e38T^{2}
71 1+4.06e19T+7.52e38T2 1 + 4.06e19T + 7.52e38T^{2}
73 1+8.70e18T+1.34e39T2 1 + 8.70e18T + 1.34e39T^{2}
79 1−3.94e19iT−7.08e39T2 1 - 3.94e19iT - 7.08e39T^{2}
83 1+5.89e19iT−1.99e40T2 1 + 5.89e19iT - 1.99e40T^{2}
89 1+3.06e20iT−8.65e40T2 1 + 3.06e20iT - 8.65e40T^{2}
97 1−8.97e20T+5.27e41T2 1 - 8.97e20T + 5.27e41T^{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

−10.86132388847769170892640132869, −10.02160760526458356722103318806, −8.777456231070154193547203639811, −7.75532249524154713911521915005, −7.08192430928741164890066569922, −5.93528957776740292308350167251, −4.39667926460702973533418718652, −3.32290696029129225600824502739, −1.75909580150700093699538524152, −0.68076613142950523384761081668, 0.00746980967300814516617844286, 1.39458874428113589378849582360, 2.46397985181050862727732812147, 3.34257488040877385429899859169, 4.89261115620107809617030952227, 6.26164745358750877098285918276, 7.41595308184708066923703852518, 8.431531706700713644663151347628, 9.192245091056178340081448671465, 10.26403454062706405550453944943

Graph of the ZZ-function along the critical line