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Properties

Label 2-2160-15.14-c2-0-45
Degree 22
Conductor 21602160
Sign 0.806+0.590i0.806 + 0.590i
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  + (−4.03 − 2.95i)5-s − 7.07i·7-s + 9.43i·11-s + 6.86i·13-s + 6.05·17-s + 14.3·19-s − 11.4·23-s + (7.55 + 23.8i)25-s − 21.2i·29-s + 8.47·31-s + (−20.8 + 28.5i)35-s − 22.0i·37-s + 58.8i·41-s + 49.4i·43-s − 49.9·47-s + ⋯
L(s)  = 1  + (−0.806 − 0.590i)5-s − 1.01i·7-s + 0.857i·11-s + 0.528i·13-s + 0.356·17-s + 0.754·19-s − 0.499·23-s + (0.302 + 0.953i)25-s − 0.731i·29-s + 0.273·31-s + (−0.596 + 0.815i)35-s − 0.595i·37-s + 1.43i·41-s + 1.14i·43-s − 1.06·47-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((0.806+0.590i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.590i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s+1) L(s)=((0.806+0.590i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.806 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: 0.806+0.590i0.806 + 0.590i
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), 0.806+0.590i)(2,\ 2160,\ (\ :1),\ 0.806 + 0.590i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.5761099341.576109934
L(12)L(\frac12) ≈\approx 1.5761099341.576109934
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+(4.03+2.95i)T 1 + (4.03 + 2.95i)T
good7 1+7.07iT−49T2 1 + 7.07iT - 49T^{2}
11 1−9.43iT−121T2 1 - 9.43iT - 121T^{2}
13 1−6.86iT−169T2 1 - 6.86iT - 169T^{2}
17 1−6.05T+289T2 1 - 6.05T + 289T^{2}
19 1−14.3T+361T2 1 - 14.3T + 361T^{2}
23 1+11.4T+529T2 1 + 11.4T + 529T^{2}
29 1+21.2iT−841T2 1 + 21.2iT - 841T^{2}
31 1−8.47T+961T2 1 - 8.47T + 961T^{2}
37 1+22.0iT−1.36e3T2 1 + 22.0iT - 1.36e3T^{2}
41 1−58.8iT−1.68e3T2 1 - 58.8iT - 1.68e3T^{2}
43 1−49.4iT−1.84e3T2 1 - 49.4iT - 1.84e3T^{2}
47 1+49.9T+2.20e3T2 1 + 49.9T + 2.20e3T^{2}
53 1−69.7T+2.80e3T2 1 - 69.7T + 2.80e3T^{2}
59 1−50.0iT−3.48e3T2 1 - 50.0iT - 3.48e3T^{2}
61 1−92.9T+3.72e3T2 1 - 92.9T + 3.72e3T^{2}
67 1+42.1iT−4.48e3T2 1 + 42.1iT - 4.48e3T^{2}
71 1−84.2iT−5.04e3T2 1 - 84.2iT - 5.04e3T^{2}
73 1+108.iT−5.32e3T2 1 + 108. iT - 5.32e3T^{2}
79 1+23.2T+6.24e3T2 1 + 23.2T + 6.24e3T^{2}
83 1+55.7T+6.88e3T2 1 + 55.7T + 6.88e3T^{2}
89 1+73.1iT−7.92e3T2 1 + 73.1iT - 7.92e3T^{2}
97 1+69.4iT−9.40e3T2 1 + 69.4iT - 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

−8.739976949290723049232254117541, −7.88808686944477107569743174457, −7.40963615252807066593592048371, −6.67274896704666854417022919805, −5.54083473880432191463416211298, −4.49693534520114803543079063565, −4.15932536083997618103564692248, −3.09926694274561557231611334517, −1.65378008342917585362840011614, −0.62611284965658869552345471706, 0.71118085802800215924709131197, 2.29908172661763065388746246672, 3.19957484482877561748838747235, 3.82947848394493303875506986097, 5.18028230131197651955438396623, 5.71628292485103925466863016434, 6.68576327157238005030849313294, 7.45050410339879816326074856229, 8.341248841939898452455308136920, 8.694884605917312469898442154010

Graph of the ZZ-function along the critical line