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Properties

Label 2-38e2-19.10-c0-0-2
Degree $2$
Conductor $1444$
Sign $0.944 + 0.327i$
Analytic cond. $0.720649$
Root an. cond. $0.848910$
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  + (0.939 + 0.342i)5-s + (0.5 − 0.866i)7-s + (0.173 − 0.984i)9-s + (0.5 + 0.866i)11-s + (−0.173 − 0.984i)17-s + (−1.87 + 0.684i)23-s + (0.766 − 0.642i)35-s + (0.939 + 0.342i)43-s + (0.5 − 0.866i)45-s + (−0.173 + 0.984i)47-s + (0.173 + 0.984i)55-s + (0.939 − 0.342i)61-s + (−0.766 − 0.642i)63-s + (−0.766 + 0.642i)73-s + 0.999·77-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)5-s + (0.5 − 0.866i)7-s + (0.173 − 0.984i)9-s + (0.5 + 0.866i)11-s + (−0.173 − 0.984i)17-s + (−1.87 + 0.684i)23-s + (0.766 − 0.642i)35-s + (0.939 + 0.342i)43-s + (0.5 − 0.866i)45-s + (−0.173 + 0.984i)47-s + (0.173 + 0.984i)55-s + (0.939 − 0.342i)61-s + (−0.766 − 0.642i)63-s + (−0.766 + 0.642i)73-s + 0.999·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $0.944 + 0.327i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ 0.944 + 0.327i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.357127133\)
\(L(\frac12)\) \(\approx\) \(1.357127133\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.173 + 0.984i)T^{2} \)
5 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (1.87 - 0.684i)T + (0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.939 - 0.342i)T^{2} \)
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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

−9.750346099088568601246495598411, −9.169139892239437205339692894111, −7.926263207608277205391561553303, −7.18301303840235298598680425120, −6.46448341307872176006544423912, −5.68296245440161214425124447464, −4.48780028805021023472260622569, −3.80771342514797373758668450991, −2.43271337586194057182518111319, −1.34147835326360555266191305555, 1.71911695996803843455250430254, 2.33897840339074086965967395856, 3.84703604604651010125958572552, 4.87090081792470773872671750546, 5.84062979180706055778506595146, 6.07859047965599710770436575432, 7.46308633260298404487336734983, 8.536953037075978065272262520588, 8.642961104448078223241792247828, 9.848508414262175129092082403631

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