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Properties

Label 2-72-72.11-c3-0-26
Degree 22
Conductor 7272
Sign 0.225+0.974i0.225 + 0.974i
Analytic cond. 4.248134.24813
Root an. cond. 2.061102.06110
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.42 − 1.45i)2-s + (1.76 − 4.88i)3-s + (3.78 − 7.04i)4-s + (4.95 + 8.58i)5-s + (−2.82 − 14.4i)6-s + (11.4 + 6.61i)7-s + (−1.04 − 22.6i)8-s + (−20.7 − 17.2i)9-s + (24.5 + 13.6i)10-s + (−23.4 − 13.5i)11-s + (−27.7 − 30.9i)12-s + (−41.1 + 23.7i)13-s + (37.3 − 0.577i)14-s + (50.7 − 9.11i)15-s + (−35.3 − 53.3i)16-s + 88.9i·17-s + ⋯
L(s)  = 1  + (0.858 − 0.513i)2-s + (0.338 − 0.940i)3-s + (0.473 − 0.881i)4-s + (0.443 + 0.768i)5-s + (−0.192 − 0.981i)6-s + (0.618 + 0.356i)7-s + (−0.0463 − 0.998i)8-s + (−0.770 − 0.637i)9-s + (0.774 + 0.431i)10-s + (−0.642 − 0.370i)11-s + (−0.668 − 0.743i)12-s + (−0.878 + 0.507i)13-s + (0.713 − 0.0110i)14-s + (0.873 − 0.156i)15-s + (−0.552 − 0.833i)16-s + 1.26i·17-s + ⋯

Functional equation

Λ(s)=(72s/2 ΓC(s) L(s)=((0.225+0.974i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(72s/2 ΓC(s+3/2) L(s)=((0.225+0.974i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7272    =    23⋅322^{3} \cdot 3^{2}
Sign: 0.225+0.974i0.225 + 0.974i
Analytic conductor: 4.248134.24813
Root analytic conductor: 2.061102.06110
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ72(11,⋅)\chi_{72} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 72, ( :3/2), 0.225+0.974i)(2,\ 72,\ (\ :3/2),\ 0.225 + 0.974i)

Particular Values

L(2)L(2) ≈\approx 2.07104−1.64656i2.07104 - 1.64656i
L(12)L(\frac12) ≈\approx 2.07104−1.64656i2.07104 - 1.64656i
L(52)L(\frac{5}{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+(−2.42+1.45i)T 1 + (-2.42 + 1.45i)T
3 1+(−1.76+4.88i)T 1 + (-1.76 + 4.88i)T
good5 1+(−4.95−8.58i)T+(−62.5+108.i)T2 1 + (-4.95 - 8.58i)T + (-62.5 + 108. i)T^{2}
7 1+(−11.4−6.61i)T+(171.5+297.i)T2 1 + (-11.4 - 6.61i)T + (171.5 + 297. i)T^{2}
11 1+(23.4+13.5i)T+(665.5+1.15e3i)T2 1 + (23.4 + 13.5i)T + (665.5 + 1.15e3i)T^{2}
13 1+(41.1−23.7i)T+(1.09e3−1.90e3i)T2 1 + (41.1 - 23.7i)T + (1.09e3 - 1.90e3i)T^{2}
17 1−88.9iT−4.91e3T2 1 - 88.9iT - 4.91e3T^{2}
19 1−88.1T+6.85e3T2 1 - 88.1T + 6.85e3T^{2}
23 1+(−89.5−155.i)T+(−6.08e3+1.05e4i)T2 1 + (-89.5 - 155. i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+(−47.7+82.7i)T+(−1.21e4−2.11e4i)T2 1 + (-47.7 + 82.7i)T + (-1.21e4 - 2.11e4i)T^{2}
31 1+(23.6−13.6i)T+(1.48e4−2.57e4i)T2 1 + (23.6 - 13.6i)T + (1.48e4 - 2.57e4i)T^{2}
37 1+117.iT−5.06e4T2 1 + 117. iT - 5.06e4T^{2}
41 1+(258.−149.i)T+(3.44e4−5.96e4i)T2 1 + (258. - 149. i)T + (3.44e4 - 5.96e4i)T^{2}
43 1+(−151.+262.i)T+(−3.97e4−6.88e4i)T2 1 + (-151. + 262. i)T + (-3.97e4 - 6.88e4i)T^{2}
47 1+(254.−440.i)T+(−5.19e4−8.99e4i)T2 1 + (254. - 440. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+415.T+1.48e5T2 1 + 415.T + 1.48e5T^{2}
59 1+(−361.+208.i)T+(1.02e5−1.77e5i)T2 1 + (-361. + 208. i)T + (1.02e5 - 1.77e5i)T^{2}
61 1+(555.+320.i)T+(1.13e5+1.96e5i)T2 1 + (555. + 320. i)T + (1.13e5 + 1.96e5i)T^{2}
67 1+(210.+364.i)T+(−1.50e5+2.60e5i)T2 1 + (210. + 364. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1−655.T+3.57e5T2 1 - 655.T + 3.57e5T^{2}
73 1−1.02e3T+3.89e5T2 1 - 1.02e3T + 3.89e5T^{2}
79 1+(503.+290.i)T+(2.46e5+4.26e5i)T2 1 + (503. + 290. i)T + (2.46e5 + 4.26e5i)T^{2}
83 1+(437.+252.i)T+(2.85e5+4.95e5i)T2 1 + (437. + 252. i)T + (2.85e5 + 4.95e5i)T^{2}
89 1+407.iT−7.04e5T2 1 + 407. iT - 7.04e5T^{2}
97 1+(479.−830.i)T+(−4.56e5−7.90e5i)T2 1 + (479. - 830. i)T + (-4.56e5 - 7.90e5i)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

−13.91247194444128297858659413744, −12.90987264495768779404400449973, −11.84573556165835793106627463306, −10.91680415587245305741505595949, −9.535476781532531355416343115649, −7.78856744671266819586010299266, −6.53564464127418003177959459776, −5.32647663256936260067794702601, −3.12058678890598239199411571828, −1.83287631348350167287001722023, 2.84776784243613069750441332941, 4.83056697294667807353039183663, 5.14933499592163271503703590476, 7.32569628652441482539556427190, 8.507612595387772481942524930114, 9.800168083618391960895809436879, 11.12831936082991839770580908140, 12.42635470097930364821674624754, 13.54889191088466792645527972223, 14.41182144736593139911086275521

Graph of the ZZ-function along the critical line