Properties

Label 2-539-7.4-c1-0-17
Degree $2$
Conductor $539$
Sign $0.968 + 0.250i$
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  + (−1.5 + 2.59i)3-s + (1 − 1.73i)4-s + (−0.5 − 0.866i)5-s + (−3 − 5.19i)9-s + (0.5 − 0.866i)11-s + (3 + 5.19i)12-s + 4·13-s + 3·15-s + (−1.99 − 3.46i)16-s + (1 − 1.73i)17-s + (−3 − 5.19i)19-s − 1.99·20-s + (2.5 + 4.33i)23-s + (2 − 3.46i)25-s + 9·27-s + ⋯
L(s)  = 1  + (−0.866 + 1.49i)3-s + (0.5 − 0.866i)4-s + (−0.223 − 0.387i)5-s + (−1 − 1.73i)9-s + (0.150 − 0.261i)11-s + (0.866 + 1.49i)12-s + 1.10·13-s + 0.774·15-s + (−0.499 − 0.866i)16-s + (0.242 − 0.420i)17-s + (−0.688 − 1.19i)19-s − 0.447·20-s + (0.521 + 0.902i)23-s + (0.400 − 0.692i)25-s + 1.73·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13016 - 0.144020i\)
\(L(\frac12)\) \(\approx\) \(1.13016 - 0.144020i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.5 - 2.59i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.5 - 4.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - T + 71T^{2} \)
73 \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3 + 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5T + 97T^{2} \)
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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

−10.81458286701414932952960157985, −10.07748303045541033945219884794, −9.274603007669605895015467172776, −8.456364407103342615200952250870, −6.72144933814625040966032359335, −6.04199127867474495332340965947, −5.04378830754702806893438670990, −4.43834649207526902337970565020, −3.07088099333140493740563833394, −0.837643650455424852969835036657, 1.38502781910885349746250986438, 2.68281552652243656909613531518, 4.03503124374891900694335864642, 5.68860159008438571200971977085, 6.60497177753532530003732619242, 6.98197256910416697071408968835, 8.091580210040817501251464947968, 8.532041915779741812335993532336, 10.46566045079068343841730802730, 11.05016640839016141404322879860

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