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Properties

Label 2-2160-15.14-c2-0-85
Degree 22
Conductor 21602160
Sign −0.354+0.934i-0.354 + 0.934i
Analytic cond. 58.855758.8557
Root an. cond. 7.671747.67174
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.67 + 1.77i)5-s − 9.67i·7-s − 8.58i·11-s − 19.0i·13-s + 26.2·17-s − 26.6·19-s + 25.8·23-s + (18.7 + 16.5i)25-s − 18.3i·29-s + 7.56·31-s + (17.1 − 45.2i)35-s + 47.0i·37-s + 0.602i·41-s − 62.2i·43-s − 79.2·47-s + ⋯
L(s)  = 1  + (0.934 + 0.354i)5-s − 1.38i·7-s − 0.780i·11-s − 1.46i·13-s + 1.54·17-s − 1.40·19-s + 1.12·23-s + (0.748 + 0.663i)25-s − 0.633i·29-s + 0.243·31-s + (0.490 − 1.29i)35-s + 1.27i·37-s + 0.0146i·41-s − 1.44i·43-s − 1.68·47-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((−0.354+0.934i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 + 0.934i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s+1) L(s)=((−0.354+0.934i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.354 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: −0.354+0.934i-0.354 + 0.934i
Analytic conductor: 58.855758.8557
Root analytic conductor: 7.671747.67174
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ2160(1889,⋅)\chi_{2160} (1889, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2160, ( :1), −0.354+0.934i)(2,\ 2160,\ (\ :1),\ -0.354 + 0.934i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.1955244752.195524475
L(12)L(\frac12) ≈\approx 2.1955244752.195524475
L(2)L(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
3 1 1
5 1+(−4.67−1.77i)T 1 + (-4.67 - 1.77i)T
good7 1+9.67iT−49T2 1 + 9.67iT - 49T^{2}
11 1+8.58iT−121T2 1 + 8.58iT - 121T^{2}
13 1+19.0iT−169T2 1 + 19.0iT - 169T^{2}
17 1−26.2T+289T2 1 - 26.2T + 289T^{2}
19 1+26.6T+361T2 1 + 26.6T + 361T^{2}
23 1−25.8T+529T2 1 - 25.8T + 529T^{2}
29 1+18.3iT−841T2 1 + 18.3iT - 841T^{2}
31 1−7.56T+961T2 1 - 7.56T + 961T^{2}
37 1−47.0iT−1.36e3T2 1 - 47.0iT - 1.36e3T^{2}
41 1−0.602iT−1.68e3T2 1 - 0.602iT - 1.68e3T^{2}
43 1+62.2iT−1.84e3T2 1 + 62.2iT - 1.84e3T^{2}
47 1+79.2T+2.20e3T2 1 + 79.2T + 2.20e3T^{2}
53 1+80.9T+2.80e3T2 1 + 80.9T + 2.80e3T^{2}
59 1+15.4iT−3.48e3T2 1 + 15.4iT - 3.48e3T^{2}
61 1+62.7T+3.72e3T2 1 + 62.7T + 3.72e3T^{2}
67 1+48.3iT−4.48e3T2 1 + 48.3iT - 4.48e3T^{2}
71 1−99.6iT−5.04e3T2 1 - 99.6iT - 5.04e3T^{2}
73 1−60.7iT−5.32e3T2 1 - 60.7iT - 5.32e3T^{2}
79 1+60.8T+6.24e3T2 1 + 60.8T + 6.24e3T^{2}
83 1−77.0T+6.88e3T2 1 - 77.0T + 6.88e3T^{2}
89 1−28.7iT−7.92e3T2 1 - 28.7iT - 7.92e3T^{2}
97 1−23.3iT−9.40e3T2 1 - 23.3iT - 9.40e3T^{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

−8.497795735563035068758299825249, −7.926277741319408507865010052865, −7.05374297536912331215295027737, −6.29313803773487121451208911586, −5.56235424960251835500338101774, −4.70402660745777426230087945774, −3.46242385943666980694305466929, −2.94778550327228898118700052001, −1.45565562617958540916285393484, −0.53342363632614774282972574390, 1.49874903102563990594638261815, 2.14048435696822535540036752973, 3.14377283839421280705011034777, 4.57618315042937533531264563492, 5.09288662030142537174071134969, 6.08744625893815798150342659796, 6.52657430694128270226452804667, 7.62038508662471202750183131588, 8.610852055610776960990192007497, 9.217288083025497852584186351115

Graph of the ZZ-function along the critical line