Properties

Label 2-2160-15.14-c2-0-5
Degree 22
Conductor 21602160
Sign 0.3540.934i0.354 - 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/2ΓC(s)L(s)=((0.3540.934i)Λ(3s)\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/2ΓC(s+1)L(s)=((0.3540.934i)Λ(1s)\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    =    243352^{4} \cdot 3^{3} \cdot 5
Sign: 0.3540.934i0.354 - 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), 0.3540.934i)(2,\ 2160,\ (\ :1),\ 0.354 - 0.934i)

Particular Values

L(32)L(\frac{3}{2}) \approx 0.36935409650.3693540965
L(12)L(\frac12) \approx 0.36935409650.3693540965
L(2)L(2) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)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+9.67iT49T2 1 + 9.67iT - 49T^{2}
11 18.58iT121T2 1 - 8.58iT - 121T^{2}
13 1+19.0iT169T2 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}
29 118.3iT841T2 1 - 18.3iT - 841T^{2}
31 17.56T+961T2 1 - 7.56T + 961T^{2}
37 147.0iT1.36e3T2 1 - 47.0iT - 1.36e3T^{2}
41 1+0.602iT1.68e3T2 1 + 0.602iT - 1.68e3T^{2}
43 1+62.2iT1.84e3T2 1 + 62.2iT - 1.84e3T^{2}
47 179.2T+2.20e3T2 1 - 79.2T + 2.20e3T^{2}
53 180.9T+2.80e3T2 1 - 80.9T + 2.80e3T^{2}
59 115.4iT3.48e3T2 1 - 15.4iT - 3.48e3T^{2}
61 1+62.7T+3.72e3T2 1 + 62.7T + 3.72e3T^{2}
67 1+48.3iT4.48e3T2 1 + 48.3iT - 4.48e3T^{2}
71 1+99.6iT5.04e3T2 1 + 99.6iT - 5.04e3T^{2}
73 160.7iT5.32e3T2 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+28.7iT7.92e3T2 1 + 28.7iT - 7.92e3T^{2}
97 123.3iT9.40e3T2 1 - 23.3iT - 9.40e3T^{2}
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   L(s)=p j=12(1αj,pps)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

−8.767481431806117153547027984923, −8.308186673216523821059431759955, −7.38112493236926714436762020224, −7.01265634261423549356667470916, −5.94452342456817243759195605395, −4.67507946075053921918137043778, −4.28625720462524095977880652932, −3.46327291747985199015098707775, −2.15377653019875470319429867648, −0.75190572412762293496804776055, 0.12566035570467189255189909390, 2.06788284377989074428568083138, 2.64761915605545553396169740500, 4.08108478622502857701766072504, 4.37565980028403929525772135351, 5.80027520943528213257954351681, 6.34399751351649491437923738059, 7.11485056684166560939632309963, 8.183976217269618576808322692661, 8.776958457061252613987336435474

Graph of the ZZ-function along the critical line