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Properties

Label 2-2888-152.123-c0-0-2
Degree $2$
Conductor $2888$
Sign $-0.0231 - 0.999i$
Analytic cond. $1.44129$
Root an. cond. $1.20054$
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 + (−1.43 + 1.20i)3-s + (−0.939 + 0.342i)4-s + (−1.43 − 1.20i)6-s + (−0.5 − 0.866i)8-s + (0.439 − 2.49i)9-s + (−0.173 − 0.300i)11-s + (0.939 − 1.62i)12-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + 2.53·18-s + (0.266 − 0.223i)22-s + (1.76 + 0.642i)24-s + (0.766 + 0.642i)25-s + (1.43 + 2.49i)27-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (−1.43 + 1.20i)3-s + (−0.939 + 0.342i)4-s + (−1.43 − 1.20i)6-s + (−0.5 − 0.866i)8-s + (0.439 − 2.49i)9-s + (−0.173 − 0.300i)11-s + (0.939 − 1.62i)12-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + 2.53·18-s + (0.266 − 0.223i)22-s + (1.76 + 0.642i)24-s + (0.766 + 0.642i)25-s + (1.43 + 2.49i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0231 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0231 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.0231 - 0.999i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2555, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ -0.0231 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6363372897\)
\(L(\frac12)\) \(\approx\) \(0.6363372897\)
\(L(1)\) 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 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
5 \( 1 + (-0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
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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

−9.361192209077222278838679276179, −8.464543364690452073840032926887, −7.33491635638480040591879259964, −6.71612597187793685216878101692, −5.89388588031251997879505290431, −5.35031217094851129526226929642, −4.72341003911381127160521761349, −4.01387328541947606907394790053, −3.10406246297355204226742237205, −0.65478739918531091125600927708, 0.905599266671765704664414227366, 1.83720065416355188228658964347, 2.74211724122067830910684810425, 4.21432614299198279305933164858, 4.86135652721886566743398130409, 5.78174463730179939181739265694, 6.23089341703450122113155051271, 7.17136436638907362084785214342, 7.969303771140136466558771894529, 8.710741924869237087602087911079

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