Properties

Label 2-160-40.19-c2-0-0
Degree 22
Conductor 160160
Sign 0.949+0.313i-0.949 + 0.313i
Analytic cond. 4.359684.35968
Root an. cond. 2.087982.08798
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  + 4.79i·3-s + (−4.35 − 2.46i)5-s − 7.67·7-s − 13.9·9-s + 0.472·11-s + 4.10·13-s + (11.7 − 20.8i)15-s − 2.26i·17-s − 26.3·19-s − 36.7i·21-s − 9.73·23-s + (12.8 + 21.4i)25-s − 23.6i·27-s + 41.6i·29-s + 22.0i·31-s + ⋯
L(s)  = 1  + 1.59i·3-s + (−0.870 − 0.492i)5-s − 1.09·7-s − 1.54·9-s + 0.0429·11-s + 0.316·13-s + (0.785 − 1.38i)15-s − 0.133i·17-s − 1.38·19-s − 1.75i·21-s − 0.423·23-s + (0.515 + 0.856i)25-s − 0.877i·27-s + 1.43i·29-s + 0.710i·31-s + ⋯

Functional equation

Λ(s)=(160s/2ΓC(s)L(s)=((0.949+0.313i)Λ(3s)\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(160s/2ΓC(s+1)L(s)=((0.949+0.313i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 160160    =    2552^{5} \cdot 5
Sign: 0.949+0.313i-0.949 + 0.313i
Analytic conductor: 4.359684.35968
Root analytic conductor: 2.087982.08798
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ160(79,)\chi_{160} (79, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 160, ( :1), 0.949+0.313i)(2,\ 160,\ (\ :1),\ -0.949 + 0.313i)

Particular Values

L(32)L(\frac{3}{2}) \approx 0.06699780.416961i0.0669978 - 0.416961i
L(12)L(\frac12) \approx 0.06699780.416961i0.0669978 - 0.416961i
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
5 1+(4.35+2.46i)T 1 + (4.35 + 2.46i)T
good3 14.79iT9T2 1 - 4.79iT - 9T^{2}
7 1+7.67T+49T2 1 + 7.67T + 49T^{2}
11 10.472T+121T2 1 - 0.472T + 121T^{2}
13 14.10T+169T2 1 - 4.10T + 169T^{2}
17 1+2.26iT289T2 1 + 2.26iT - 289T^{2}
19 1+26.3T+361T2 1 + 26.3T + 361T^{2}
23 1+9.73T+529T2 1 + 9.73T + 529T^{2}
29 141.6iT841T2 1 - 41.6iT - 841T^{2}
31 122.0iT961T2 1 - 22.0iT - 961T^{2}
37 1+51.7T+1.36e3T2 1 + 51.7T + 1.36e3T^{2}
41 115.0T+1.68e3T2 1 - 15.0T + 1.68e3T^{2}
43 19.31iT1.84e3T2 1 - 9.31iT - 1.84e3T^{2}
47 1+8.76T+2.20e3T2 1 + 8.76T + 2.20e3T^{2}
53 139.9T+2.80e3T2 1 - 39.9T + 2.80e3T^{2}
59 177.1T+3.48e3T2 1 - 77.1T + 3.48e3T^{2}
61 114.7iT3.72e3T2 1 - 14.7iT - 3.72e3T^{2}
67 1+75.8iT4.48e3T2 1 + 75.8iT - 4.48e3T^{2}
71 181.0iT5.04e3T2 1 - 81.0iT - 5.04e3T^{2}
73 183.4iT5.32e3T2 1 - 83.4iT - 5.32e3T^{2}
79 1+100.iT6.24e3T2 1 + 100. iT - 6.24e3T^{2}
83 1+0.266iT6.88e3T2 1 + 0.266iT - 6.88e3T^{2}
89 185.5T+7.92e3T2 1 - 85.5T + 7.92e3T^{2}
97 199.1iT9.40e3T2 1 - 99.1iT - 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

−13.05328635291272971641202443112, −12.15190037070252163789830133682, −10.94408528717801768996126414159, −10.23314446510900431782984038463, −9.140797653733188351427275684040, −8.479577009065566716211189839671, −6.78108835037866087607389921621, −5.29686078788369774286768562172, −4.14891265575917097426570443984, −3.32190675510560422555138442292, 0.25201815960446320434649560492, 2.37135386090958401251202593006, 3.84169213511014197246134519234, 6.12503656842603105568442048196, 6.75100669789164617622683493149, 7.74227072789023623619549216181, 8.646773809056524844614992209307, 10.21217482539657267923065126312, 11.42267679623298232955151169418, 12.26156590532806190951887762405

Graph of the ZZ-function along the critical line