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Properties

Label 2-272-17.4-c3-0-17
Degree 22
Conductor 272272
Sign 0.999+0.0318i0.999 + 0.0318i
Analytic cond. 16.048516.0485
Root an. cond. 4.006064.00606
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  + (0.657 − 0.657i)3-s + (13.8 − 13.8i)5-s + (14.0 + 14.0i)7-s + 26.1i·9-s + (29.5 + 29.5i)11-s − 19.0·13-s − 18.1i·15-s + (44.8 − 53.8i)17-s + 149. i·19-s + 18.4·21-s + (−80.3 − 80.3i)23-s − 257. i·25-s + (34.9 + 34.9i)27-s + (104. − 104. i)29-s + (73.6 − 73.6i)31-s + ⋯
L(s)  = 1  + (0.126 − 0.126i)3-s + (1.23 − 1.23i)5-s + (0.756 + 0.756i)7-s + 0.968i·9-s + (0.808 + 0.808i)11-s − 0.406·13-s − 0.312i·15-s + (0.640 − 0.768i)17-s + 1.80i·19-s + 0.191·21-s + (−0.728 − 0.728i)23-s − 2.06i·25-s + (0.248 + 0.248i)27-s + (0.666 − 0.666i)29-s + (0.426 − 0.426i)31-s + ⋯

Functional equation

Λ(s)=(272s/2 ΓC(s) L(s)=((0.999+0.0318i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0318i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(272s/2 ΓC(s+3/2) L(s)=((0.999+0.0318i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0318i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 272272    =    24⋅172^{4} \cdot 17
Sign: 0.999+0.0318i0.999 + 0.0318i
Analytic conductor: 16.048516.0485
Root analytic conductor: 4.006064.00606
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ272(225,⋅)\chi_{272} (225, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 272, ( :3/2), 0.999+0.0318i)(2,\ 272,\ (\ :3/2),\ 0.999 + 0.0318i)

Particular Values

L(2)L(2) ≈\approx 2.6476455752.647645575
L(12)L(\frac12) ≈\approx 2.6476455752.647645575
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 1
17 1+(−44.8+53.8i)T 1 + (-44.8 + 53.8i)T
good3 1+(−0.657+0.657i)T−27iT2 1 + (-0.657 + 0.657i)T - 27iT^{2}
5 1+(−13.8+13.8i)T−125iT2 1 + (-13.8 + 13.8i)T - 125iT^{2}
7 1+(−14.0−14.0i)T+343iT2 1 + (-14.0 - 14.0i)T + 343iT^{2}
11 1+(−29.5−29.5i)T+1.33e3iT2 1 + (-29.5 - 29.5i)T + 1.33e3iT^{2}
13 1+19.0T+2.19e3T2 1 + 19.0T + 2.19e3T^{2}
19 1−149.iT−6.85e3T2 1 - 149. iT - 6.85e3T^{2}
23 1+(80.3+80.3i)T+1.21e4iT2 1 + (80.3 + 80.3i)T + 1.21e4iT^{2}
29 1+(−104.+104.i)T−2.43e4iT2 1 + (-104. + 104. i)T - 2.43e4iT^{2}
31 1+(−73.6+73.6i)T−2.97e4iT2 1 + (-73.6 + 73.6i)T - 2.97e4iT^{2}
37 1+(196.−196.i)T−5.06e4iT2 1 + (196. - 196. i)T - 5.06e4iT^{2}
41 1+(−283.−283.i)T+6.89e4iT2 1 + (-283. - 283. i)T + 6.89e4iT^{2}
43 1+20.5iT−7.95e4T2 1 + 20.5iT - 7.95e4T^{2}
47 1+81.5T+1.03e5T2 1 + 81.5T + 1.03e5T^{2}
53 1+626.iT−1.48e5T2 1 + 626. iT - 1.48e5T^{2}
59 1+301.iT−2.05e5T2 1 + 301. iT - 2.05e5T^{2}
61 1+(−352.−352.i)T+2.26e5iT2 1 + (-352. - 352. i)T + 2.26e5iT^{2}
67 1+924.T+3.00e5T2 1 + 924.T + 3.00e5T^{2}
71 1+(−748.+748.i)T−3.57e5iT2 1 + (-748. + 748. i)T - 3.57e5iT^{2}
73 1+(−4.67+4.67i)T−3.89e5iT2 1 + (-4.67 + 4.67i)T - 3.89e5iT^{2}
79 1+(364.+364.i)T+4.93e5iT2 1 + (364. + 364. i)T + 4.93e5iT^{2}
83 1+772.iT−5.71e5T2 1 + 772. iT - 5.71e5T^{2}
89 1+221.T+7.04e5T2 1 + 221.T + 7.04e5T^{2}
97 1+(7.27−7.27i)T−9.12e5iT2 1 + (7.27 - 7.27i)T - 9.12e5iT^{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

−11.85930711360463764216067533239, −10.14433648477064291932772459617, −9.665786563668854648130252129004, −8.521394242134600809584403431141, −7.87888190689127190631406560990, −6.22688199571967314616030342396, −5.25887172065036425742830833436, −4.54936648082913044662101550406, −2.23948073979511915409729156979, −1.47821202507680714715295706679, 1.23671652017196250041802484227, 2.80282391017537379112513308059, 3.95541285928726700975019740421, 5.58277177154807319796848440772, 6.54900346259609890315635030586, 7.29268206167097205292432117110, 8.801926635821145098300520220341, 9.646084239398719576991312903826, 10.59193171117278571515332174176, 11.18830622914006511473709318425

Graph of the ZZ-function along the critical line