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Properties

Label 2-464-4.3-c2-0-0
Degree 22
Conductor 464464
Sign −0.5−0.866i-0.5 - 0.866i
Analytic cond. 12.643012.6430
Root an. cond. 3.555713.55571
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  − 3.94i·3-s − 0.459·5-s + 4.40i·7-s − 6.58·9-s + 4.56i·11-s − 24.0·13-s + 1.81i·15-s − 19.4·17-s + 0.419i·19-s + 17.4·21-s + 3.76i·23-s − 24.7·25-s − 9.53i·27-s − 5.38·29-s + 40.1i·31-s + ⋯
L(s)  = 1  − 1.31i·3-s − 0.0918·5-s + 0.629i·7-s − 0.731·9-s + 0.415i·11-s − 1.85·13-s + 0.120i·15-s − 1.14·17-s + 0.0220i·19-s + 0.828·21-s + 0.163i·23-s − 0.991·25-s − 0.353i·27-s − 0.185·29-s + 1.29i·31-s + ⋯

Functional equation

Λ(s)=(464s/2 ΓC(s) L(s)=((−0.5−0.866i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(464s/2 ΓC(s+1) L(s)=((−0.5−0.866i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 464464    =    24⋅292^{4} \cdot 29
Sign: −0.5−0.866i-0.5 - 0.866i
Analytic conductor: 12.643012.6430
Root analytic conductor: 3.555713.55571
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ464(175,⋅)\chi_{464} (175, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 464, ( :1), −0.5−0.866i)(2,\ 464,\ (\ :1),\ -0.5 - 0.866i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.11573829290.1157382929
L(12)L(\frac12) ≈\approx 0.11573829290.1157382929
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
29 1+5.38T 1 + 5.38T
good3 1+3.94iT−9T2 1 + 3.94iT - 9T^{2}
5 1+0.459T+25T2 1 + 0.459T + 25T^{2}
7 1−4.40iT−49T2 1 - 4.40iT - 49T^{2}
11 1−4.56iT−121T2 1 - 4.56iT - 121T^{2}
13 1+24.0T+169T2 1 + 24.0T + 169T^{2}
17 1+19.4T+289T2 1 + 19.4T + 289T^{2}
19 1−0.419iT−361T2 1 - 0.419iT - 361T^{2}
23 1−3.76iT−529T2 1 - 3.76iT - 529T^{2}
31 1−40.1iT−961T2 1 - 40.1iT - 961T^{2}
37 1−4.71T+1.36e3T2 1 - 4.71T + 1.36e3T^{2}
41 1+19.4T+1.68e3T2 1 + 19.4T + 1.68e3T^{2}
43 1+3.64iT−1.84e3T2 1 + 3.64iT - 1.84e3T^{2}
47 1−38.3iT−2.20e3T2 1 - 38.3iT - 2.20e3T^{2}
53 1−36.7T+2.80e3T2 1 - 36.7T + 2.80e3T^{2}
59 1+54.8iT−3.48e3T2 1 + 54.8iT - 3.48e3T^{2}
61 1+19.5T+3.72e3T2 1 + 19.5T + 3.72e3T^{2}
67 1+11.7iT−4.48e3T2 1 + 11.7iT - 4.48e3T^{2}
71 1−129.iT−5.04e3T2 1 - 129. iT - 5.04e3T^{2}
73 1+16.7T+5.32e3T2 1 + 16.7T + 5.32e3T^{2}
79 1+94.2iT−6.24e3T2 1 + 94.2iT - 6.24e3T^{2}
83 1+74.1iT−6.88e3T2 1 + 74.1iT - 6.88e3T^{2}
89 1+127.T+7.92e3T2 1 + 127.T + 7.92e3T^{2}
97 1+75.2T+9.40e3T2 1 + 75.2T + 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

−11.46870917112216582126204905406, −10.17097475222271038872376204039, −9.260349058373212448190295909709, −8.235522646849581981260415044233, −7.31626699283133251375545179302, −6.77995481867770190426178370573, −5.60155467435681173003611810247, −4.49388782881884100149994128977, −2.66166443237625149160999668865, −1.82382005473247197071114963843, 0.04380652457935179408458876191, 2.44866929941052654764901215276, 3.87545995466982188122926298232, 4.54209940198768188272414933225, 5.51340433627426580674298251445, 6.89364055002179592712454788916, 7.81689903200411543766993108680, 9.029650500481842504018777384250, 9.759680890826741042445824533860, 10.38842010475376215993569927036

Graph of the ZZ-function along the critical line