Properties

Label 2-38e2-76.7-c0-0-2
Degree $2$
Conductor $1444$
Sign $-0.813 - 0.582i$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.809 + 1.40i)5-s − 0.999·8-s + (−0.5 + 0.866i)9-s + (−0.809 + 1.40i)10-s + (0.309 − 0.535i)13-s + (−0.5 − 0.866i)16-s + (−0.309 − 0.535i)17-s − 0.999·18-s − 1.61·20-s + (−0.809 + 1.40i)25-s + 0.618·26-s + (0.309 − 0.535i)29-s + (0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.809 + 1.40i)5-s − 0.999·8-s + (−0.5 + 0.866i)9-s + (−0.809 + 1.40i)10-s + (0.309 − 0.535i)13-s + (−0.5 − 0.866i)16-s + (−0.309 − 0.535i)17-s − 0.999·18-s − 1.61·20-s + (−0.809 + 1.40i)25-s + 0.618·26-s + (0.309 − 0.535i)29-s + (0.499 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $-0.813 - 0.582i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (1375, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ -0.813 - 0.582i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.404502299\)
\(L(\frac12)\) \(\approx\) \(1.404502299\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 \)
good3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-0.309 + 0.535i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.309 + 0.535i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.61T + T^{2} \)
41 \( 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.809 - 1.40i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.809 - 1.40i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.309 - 0.535i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.02921356324748668783635519434, −9.158085645503479194147470936820, −8.196088965271123886329374651940, −7.45671796849103379695164778800, −6.73538005437345751758201343087, −5.92723310299465743909699425543, −5.38805298267408169478289116528, −4.21748794561492872211823219891, −2.99956484480367395327461808777, −2.41728951275332681582028445135, 1.03943504847659134295605049928, 2.03450429590059934495416569399, 3.31803663528528660920241828714, 4.35901754345373052383222637674, 5.04530782907051676636035405702, 5.98465889879171294376910130071, 6.47860211862361697199948645627, 8.218539493640330387860478728285, 8.900896730662888058079770393990, 9.418043779584818885541223978025

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