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Properties

Label 2-2160-15.14-c2-0-9
Degree 22
Conductor 21602160
Sign −0.872−0.488i-0.872 - 0.488i
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  + (4.36 + 2.44i)5-s − 2.79i·7-s + 18.1i·11-s − 23.0i·13-s − 5.72·17-s − 23.1·19-s + 0.271·23-s + (13.0 + 21.2i)25-s + 39.7i·29-s − 47.3·31-s + (6.82 − 12.2i)35-s − 34.8i·37-s + 13.2i·41-s + 46.7i·43-s − 40.9·47-s + ⋯
L(s)  = 1  + (0.872 + 0.488i)5-s − 0.399i·7-s + 1.64i·11-s − 1.77i·13-s − 0.336·17-s − 1.22·19-s + 0.0118·23-s + (0.523 + 0.851i)25-s + 1.37i·29-s − 1.52·31-s + (0.194 − 0.348i)35-s − 0.942i·37-s + 0.323i·41-s + 1.08i·43-s − 0.870·47-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((−0.872−0.488i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s+1) L(s)=((−0.872−0.488i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: −0.872−0.488i-0.872 - 0.488i
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), −0.872−0.488i)(2,\ 2160,\ (\ :1),\ -0.872 - 0.488i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.84967882250.8496788225
L(12)L(\frac12) ≈\approx 0.84967882250.8496788225
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+(−4.36−2.44i)T 1 + (-4.36 - 2.44i)T
good7 1+2.79iT−49T2 1 + 2.79iT - 49T^{2}
11 1−18.1iT−121T2 1 - 18.1iT - 121T^{2}
13 1+23.0iT−169T2 1 + 23.0iT - 169T^{2}
17 1+5.72T+289T2 1 + 5.72T + 289T^{2}
19 1+23.1T+361T2 1 + 23.1T + 361T^{2}
23 1−0.271T+529T2 1 - 0.271T + 529T^{2}
29 1−39.7iT−841T2 1 - 39.7iT - 841T^{2}
31 1+47.3T+961T2 1 + 47.3T + 961T^{2}
37 1+34.8iT−1.36e3T2 1 + 34.8iT - 1.36e3T^{2}
41 1−13.2iT−1.68e3T2 1 - 13.2iT - 1.68e3T^{2}
43 1−46.7iT−1.84e3T2 1 - 46.7iT - 1.84e3T^{2}
47 1+40.9T+2.20e3T2 1 + 40.9T + 2.20e3T^{2}
53 1+91.3T+2.80e3T2 1 + 91.3T + 2.80e3T^{2}
59 1−78.8iT−3.48e3T2 1 - 78.8iT - 3.48e3T^{2}
61 1−31.1T+3.72e3T2 1 - 31.1T + 3.72e3T^{2}
67 1−6.91iT−4.48e3T2 1 - 6.91iT - 4.48e3T^{2}
71 1+81.5iT−5.04e3T2 1 + 81.5iT - 5.04e3T^{2}
73 1−106.iT−5.32e3T2 1 - 106. iT - 5.32e3T^{2}
79 1+63.5T+6.24e3T2 1 + 63.5T + 6.24e3T^{2}
83 1+0.284T+6.88e3T2 1 + 0.284T + 6.88e3T^{2}
89 1−28.5iT−7.92e3T2 1 - 28.5iT - 7.92e3T^{2}
97 1+92.9iT−9.40e3T2 1 + 92.9iT - 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

−9.387693470184420087427558469301, −8.458272995796831702604062029001, −7.45469817745950880549964314735, −7.01166306414841305364930543252, −6.07213391360948891186447392939, −5.28446453240214151900805790271, −4.46434148931892996586207064854, −3.33450483032914835040803969372, −2.37766732125980999550645154237, −1.47463515624593381083772208146, 0.18879155904879151652914180847, 1.65543679345467919426987487368, 2.37979568795128069548292038516, 3.66835289118221716613950145951, 4.55443008358671759128392437993, 5.49380056410662025556531619293, 6.25463217206279760292280676176, 6.68297579499378639620142123779, 8.036487796733396788376215275443, 8.808422448399783188350318927525

Graph of the ZZ-function along the critical line