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Properties

Label 2-539-77.76-c1-0-35
Degree $2$
Conductor $539$
Sign $0.377 - 0.925i$
Analytic cond. $4.30393$
Root an. cond. $2.07459$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.60i·2-s − 2.82i·3-s − 4.79·4-s − 1.84i·5-s − 7.35·6-s + 7.29i·8-s − 4.95·9-s − 4.81·10-s + (−0.112 − 3.31i)11-s + 13.5i·12-s + 4.27·13-s − 5.21·15-s + 9.42·16-s + 3.30·17-s + 12.9i·18-s + 3.58·19-s + ⋯
L(s)  = 1  − 1.84i·2-s − 1.62i·3-s − 2.39·4-s − 0.826i·5-s − 3.00·6-s + 2.57i·8-s − 1.65·9-s − 1.52·10-s + (−0.0338 − 0.999i)11-s + 3.90i·12-s + 1.18·13-s − 1.34·15-s + 2.35·16-s + 0.801·17-s + 3.04i·18-s + 0.823·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.377 - 0.925i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ 0.377 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.980534 + 0.658808i\)
\(L(\frac12)\) \(\approx\) \(0.980534 + 0.658808i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 + (0.112 + 3.31i)T \)
good2 \( 1 + 2.60iT - 2T^{2} \)
3 \( 1 + 2.82iT - 3T^{2} \)
5 \( 1 + 1.84iT - 5T^{2} \)
13 \( 1 - 4.27T + 13T^{2} \)
17 \( 1 - 3.30T + 17T^{2} \)
19 \( 1 - 3.58T + 19T^{2} \)
23 \( 1 + 4.98T + 23T^{2} \)
29 \( 1 + 3.27iT - 29T^{2} \)
31 \( 1 - 1.58iT - 31T^{2} \)
37 \( 1 - 0.946T + 37T^{2} \)
41 \( 1 - 1.77T + 41T^{2} \)
43 \( 1 - 12.0iT - 43T^{2} \)
47 \( 1 + 8.15iT - 47T^{2} \)
53 \( 1 + 0.871T + 53T^{2} \)
59 \( 1 - 6.68iT - 59T^{2} \)
61 \( 1 - 4.67T + 61T^{2} \)
67 \( 1 + 8.66T + 67T^{2} \)
71 \( 1 - 7.09T + 71T^{2} \)
73 \( 1 - 9.35T + 73T^{2} \)
79 \( 1 - 6.48iT - 79T^{2} \)
83 \( 1 + 8.55T + 83T^{2} \)
89 \( 1 + 7.58iT - 89T^{2} \)
97 \( 1 + 3.12iT - 97T^{2} \)
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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

−10.34060734563781747548187020605, −9.235218227060918323511534522275, −8.421714759954723085466407945837, −7.88163697797247976289624691507, −6.23084963518828277282696146656, −5.31655806540424765228794646814, −3.83024016363060380246650542163, −2.78201011801063916208511098582, −1.44208191694005395484336914632, −0.806129227818929132313499884575, 3.40613438259529985905658997232, 4.21613900719244217718628015541, 5.20829239985996358663903348880, 5.95655978667186413899606964691, 6.97294227153058632252929848644, 7.890333680014755928884502666672, 8.839454079634016783840556822314, 9.652731255078676883116123722879, 10.23272055120232137442867005410, 11.16658797151685195229463581436

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