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Properties

Label 2-1539-19.18-c0-0-2
Degree $2$
Conductor $1539$
Sign $-i$
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  + i·2-s + 5-s + 7-s + i·8-s + i·10-s − 11-s + i·13-s + i·14-s − 16-s − i·19-s − i·22-s − 23-s − 26-s − i·29-s − i·31-s + ⋯
L(s)  = 1  + i·2-s + 5-s + 7-s + i·8-s + i·10-s − 11-s + i·13-s + i·14-s − 16-s − i·19-s − i·22-s − 23-s − 26-s − i·29-s − i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1539\)    =    \(3^{4} \cdot 19\)
Sign: $-i$
Analytic conductor: \(0.768061\)
Root analytic conductor: \(0.876390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1539} (892, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1539,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.505605893\)
\(L(\frac12)\) \(\approx\) \(1.505605893\)
\(L(1)\) 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 \( 1 + iT \)
good2 \( 1 - iT - T^{2} \)
5 \( 1 - T + T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 - iT - T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - iT - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + iT - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT - T^{2} \)
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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

−9.656883299494052783065525492782, −8.939822321576115238436507285482, −7.931409685147906463448897509363, −7.59882340576138892270545793225, −6.45956311165182238866042229881, −5.92582804691262182445718514209, −5.09201535370790650163208335709, −4.36972547509940045346228038538, −2.50046753692131870060783957995, −1.92056654000551664774979599775, 1.39598526579720145480991906165, 2.20964697324240044925406340760, 3.09523048201631533229628724228, 4.24741045133763046753741138707, 5.43257425761806037616478071802, 5.86845016585359178143083629459, 7.16390636250963708560097389685, 7.914641630721072843438982510575, 8.771359059351679718477926816288, 9.809169839768427480760488778256

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