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Properties

Label 2-1100-11.10-c2-0-35
Degree 22
Conductor 11001100
Sign −0.256+0.966i-0.256 + 0.966i
Analytic cond. 29.972829.9728
Root an. cond. 5.474745.47474
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  + 3.41·3-s − 0.558i·7-s + 2.66·9-s + (2.82 − 10.6i)11-s − 18.1i·13-s − 32.0i·17-s + 15.7i·19-s − 1.90i·21-s − 31.7·23-s − 21.6·27-s + 48.3i·29-s − 43.3·31-s + (9.63 − 36.3i)33-s − 21.4·37-s − 61.8i·39-s + ⋯
L(s)  = 1  + 1.13·3-s − 0.0798i·7-s + 0.296·9-s + (0.256 − 0.966i)11-s − 1.39i·13-s − 1.88i·17-s + 0.828i·19-s − 0.0908i·21-s − 1.38·23-s − 0.801·27-s + 1.66i·29-s − 1.39·31-s + (0.291 − 1.10i)33-s − 0.579·37-s − 1.58i·39-s + ⋯

Functional equation

Λ(s)=(1100s/2 ΓC(s) L(s)=((−0.256+0.966i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(1100s/2 ΓC(s+1) L(s)=((−0.256+0.966i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11001100    =    22⋅52⋅112^{2} \cdot 5^{2} \cdot 11
Sign: −0.256+0.966i-0.256 + 0.966i
Analytic conductor: 29.972829.9728
Root analytic conductor: 5.474745.47474
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ1100(901,⋅)\chi_{1100} (901, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1100, ( :1), −0.256+0.966i)(2,\ 1100,\ (\ :1),\ -0.256 + 0.966i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.1019844542.101984454
L(12)L(\frac12) ≈\approx 2.1019844542.101984454
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
5 1 1
11 1+(−2.82+10.6i)T 1 + (-2.82 + 10.6i)T
good3 1−3.41T+9T2 1 - 3.41T + 9T^{2}
7 1+0.558iT−49T2 1 + 0.558iT - 49T^{2}
13 1+18.1iT−169T2 1 + 18.1iT - 169T^{2}
17 1+32.0iT−289T2 1 + 32.0iT - 289T^{2}
19 1−15.7iT−361T2 1 - 15.7iT - 361T^{2}
23 1+31.7T+529T2 1 + 31.7T + 529T^{2}
29 1−48.3iT−841T2 1 - 48.3iT - 841T^{2}
31 1+43.3T+961T2 1 + 43.3T + 961T^{2}
37 1+21.4T+1.36e3T2 1 + 21.4T + 1.36e3T^{2}
41 1+38.9iT−1.68e3T2 1 + 38.9iT - 1.68e3T^{2}
43 1+23.3iT−1.84e3T2 1 + 23.3iT - 1.84e3T^{2}
47 1−75.2T+2.20e3T2 1 - 75.2T + 2.20e3T^{2}
53 1−8.43T+2.80e3T2 1 - 8.43T + 2.80e3T^{2}
59 1+29.6T+3.48e3T2 1 + 29.6T + 3.48e3T^{2}
61 1+64.1iT−3.72e3T2 1 + 64.1iT - 3.72e3T^{2}
67 1+18.8T+4.48e3T2 1 + 18.8T + 4.48e3T^{2}
71 1−94.8T+5.04e3T2 1 - 94.8T + 5.04e3T^{2}
73 1−0.945iT−5.32e3T2 1 - 0.945iT - 5.32e3T^{2}
79 1+73.2iT−6.24e3T2 1 + 73.2iT - 6.24e3T^{2}
83 1+161.iT−6.88e3T2 1 + 161. iT - 6.88e3T^{2}
89 1−91.5T+7.92e3T2 1 - 91.5T + 7.92e3T^{2}
97 1−33.1T+9.40e3T2 1 - 33.1T + 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

−9.118601144541406062123364267158, −8.756393943036475269173879836241, −7.76887281247716116082907075994, −7.28858885534177793606740217578, −5.88513750525548452442891740874, −5.22510329761939024673343104199, −3.67545400396807706891286091046, −3.21187951710206860130024400685, −2.10995734993204871729694823716, −0.50832468429743735143525845271, 1.79153770688133166670528255821, 2.37669566422374290195358269679, 3.91236070077916206465907960992, 4.22215916989879522826921432501, 5.76403489352851384325915939907, 6.65439771406830534507041116318, 7.59273968053250140447774575726, 8.310735605166026299840166279332, 9.111595426018866479424576894087, 9.622080100084117902951907364785

Graph of the ZZ-function along the critical line