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Properties

Label 2-1539-171.140-c0-0-0
Degree $2$
Conductor $1539$
Sign $0.999 + 0.0383i$
Analytic cond. $0.768061$
Root an. cond. $0.876390$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)7-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s + 19-s + 25-s + (0.499 − 0.866i)28-s + (0.5 − 0.866i)31-s − 37-s + (0.5 − 0.866i)43-s + (0.499 − 0.866i)52-s − 61-s + 0.999·64-s + (0.5 + 0.866i)67-s + (0.5 + 0.866i)73-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)7-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s + 19-s + 25-s + (0.499 − 0.866i)28-s + (0.5 − 0.866i)31-s − 37-s + (0.5 − 0.866i)43-s + (0.499 − 0.866i)52-s − 61-s + 0.999·64-s + (0.5 + 0.866i)67-s + (0.5 + 0.866i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1539\)    =    \(3^{4} \cdot 19\)
Sign: $0.999 + 0.0383i$
Analytic conductor: \(0.768061\)
Root analytic conductor: \(0.876390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1539} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1539,\ (\ :0),\ 0.999 + 0.0383i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.080924221\)
\(L(\frac12)\) \(\approx\) \(1.080924221\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 - T \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.498820869443401071869200797126, −8.930049294431894290673833486777, −8.312841096688338056052376035286, −7.14780447582695294703046623662, −6.25167544764299354818661998096, −5.45884459273746572262406937792, −4.82462551101106316752878679549, −3.80936670065825329478914242462, −2.40440856173645691182839201906, −1.30346006937395725879944143999, 1.12656888326601167268129999208, 2.91169442020472748113310662599, 3.62598008450832854572115211491, 4.61906893374022407338417441227, 5.30639795202839473766471746115, 6.59428820174605517888348280638, 7.45854974343366795451643102798, 8.001306040419517469551993195392, 8.740307741070049091456945951617, 9.561817816199146173121503217129

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