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Properties

Label 2-38e2-76.55-c0-0-3
Degree $2$
Conductor $1444$
Sign $0.0803 + 0.996i$
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  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (1.52 − 0.553i)5-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.280 − 1.59i)10-s + (0.473 − 0.397i)13-s + (0.766 + 0.642i)16-s + (0.107 − 0.608i)17-s + 0.999·18-s − 1.61·20-s + (1.23 − 1.04i)25-s + (−0.309 − 0.535i)26-s + (0.107 + 0.608i)29-s + (0.766 − 0.642i)32-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (1.52 − 0.553i)5-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.280 − 1.59i)10-s + (0.473 − 0.397i)13-s + (0.766 + 0.642i)16-s + (0.107 − 0.608i)17-s + 0.999·18-s − 1.61·20-s + (1.23 − 1.04i)25-s + (−0.309 − 0.535i)26-s + (0.107 + 0.608i)29-s + (0.766 − 0.642i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0803 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0803 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $0.0803 + 0.996i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (967, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ 0.0803 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.407857448\)
\(L(\frac12)\) \(\approx\) \(1.407857448\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 \)
good3 \( 1 + (-0.173 - 0.984i)T^{2} \)
5 \( 1 + (-1.52 + 0.553i)T + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.473 + 0.397i)T + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (-0.107 + 0.608i)T + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (-0.107 - 0.608i)T + (-0.939 + 0.342i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 1.61T + T^{2} \)
41 \( 1 + (1.23 + 1.04i)T + (0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 - 0.342i)T^{2} \)
53 \( 1 + (-1.52 - 0.553i)T + (0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (0.580 + 0.211i)T + (0.766 + 0.642i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (1.23 + 1.04i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (1.23 - 1.04i)T + (0.173 - 0.984i)T^{2} \)
97 \( 1 + (-0.107 + 0.608i)T + (-0.939 - 0.342i)T^{2} \)
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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

−9.675875548691937862596240136366, −8.928837718678830303623830191847, −8.329232881021267355563007396293, −7.05938258033439809873888616792, −5.80257996765357359350510009703, −5.30689593328121836199162576529, −4.58093029488703643480055333547, −3.22858442803111112757456360912, −2.18416526293137235012767347832, −1.39483876637080032541160855253, 1.61157676859970763923787486350, 3.09880078526098251944154280990, 4.03521572672376773363123808361, 5.24699278973242436061145163777, 5.99289648906688917028425128514, 6.56298543212547508019104712483, 7.12807703109162001351652306277, 8.455597315312432420033846128301, 8.972634448799396606618568951220, 9.977488424115956492959166195180

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