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Properties

Label 2-160-5.4-c3-0-6
Degree $2$
Conductor $160$
Sign $-i$
Analytic cond. $9.44030$
Root an. cond. $3.07250$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.76i·3-s + 11.1·5-s + 15.5i·7-s + 4.30·9-s + 53.2i·15-s − 74.3·21-s + 207. i·23-s + 125.·25-s + 149. i·27-s − 306·29-s + 174. i·35-s + 460.·41-s − 30.9i·43-s + 48.1·45-s − 643. i·47-s + ⋯
L(s)  = 1  + 0.916i·3-s + 0.999·5-s + 0.842i·7-s + 0.159·9-s + 0.916i·15-s − 0.772·21-s + 1.88i·23-s + 1.00·25-s + 1.06i·27-s − 1.95·29-s + 0.842i·35-s + 1.75·41-s − 0.109i·43-s + 0.159·45-s − 1.99i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-i$
Analytic conductor: \(9.44030\)
Root analytic conductor: \(3.07250\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :3/2),\ -i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.38631 + 1.38631i\)
\(L(\frac12)\) \(\approx\) \(1.38631 + 1.38631i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - 11.1T \)
good3 \( 1 - 4.76iT - 27T^{2} \)
7 \( 1 - 15.5iT - 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 - 4.91e3T^{2} \)
19 \( 1 + 6.85e3T^{2} \)
23 \( 1 - 207. iT - 1.21e4T^{2} \)
29 \( 1 + 306T + 2.43e4T^{2} \)
31 \( 1 + 2.97e4T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 - 460.T + 6.89e4T^{2} \)
43 \( 1 + 30.9iT - 7.95e4T^{2} \)
47 \( 1 + 643. iT - 1.03e5T^{2} \)
53 \( 1 - 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 - 40.2T + 2.26e5T^{2} \)
67 \( 1 + 1.09e3iT - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 3.89e5T^{2} \)
79 \( 1 + 4.93e5T^{2} \)
83 \( 1 + 1.14e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.38e3T + 7.04e5T^{2} \)
97 \( 1 - 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.78374071863924259370828546915, −11.52826403336226378199576198281, −10.50925174309259090151431033594, −9.488086514179841990883361147101, −9.084321433929143711189704304566, −7.42208168823084292373623505920, −5.89157886146161490282339699735, −5.13166426923616119496647256907, −3.59059949057182386486378014440, −1.93666941953532595376525967970, 0.997949833365966833715204881377, 2.36667406218455127757591975495, 4.32361141496976920208105231329, 5.90699923915792669497212227410, 6.86068672533443588908611754775, 7.76447838998563013933559703178, 9.142440889879378783652296202863, 10.19884772760449632941826778256, 11.08388398729350156450138184318, 12.61289108597088711463851000237

Graph of the $Z$-function along the critical line