Properties

Label 2-2888-152.43-c0-0-4
Degree $2$
Conductor $2888$
Sign $-0.392 + 0.919i$
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  + (−0.939 + 0.342i)2-s + (0.266 − 1.50i)3-s + (0.766 − 0.642i)4-s + (0.266 + 1.50i)6-s + (−0.500 + 0.866i)8-s + (−1.26 − 0.460i)9-s + (0.939 − 1.62i)11-s + (−0.766 − 1.32i)12-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + 1.34·18-s + (−0.326 + 1.85i)22-s + (1.17 + 0.984i)24-s + (0.173 + 0.984i)25-s + (−0.266 + 0.460i)27-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.266 − 1.50i)3-s + (0.766 − 0.642i)4-s + (0.266 + 1.50i)6-s + (−0.500 + 0.866i)8-s + (−1.26 − 0.460i)9-s + (0.939 − 1.62i)11-s + (−0.766 − 1.32i)12-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + 1.34·18-s + (−0.326 + 1.85i)22-s + (1.17 + 0.984i)24-s + (0.173 + 0.984i)25-s + (−0.266 + 0.460i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.392 + 0.919i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (1867, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ -0.392 + 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9263767713\)
\(L(\frac12)\) \(\approx\) \(0.9263767713\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 \)
good3 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
5 \( 1 + (-0.173 - 0.984i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.939 - 0.342i)T^{2} \)
17 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
43 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 + (-0.173 + 0.984i)T^{2} \)
59 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.173 + 0.984i)T^{2} \)
67 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
97 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
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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

−8.544124621465343456033889577869, −8.011217023921819170900712974774, −7.24148987946682066226870050408, −6.69929505224733914017651950856, −5.97160083327572019165049078227, −5.36156064732578856847028743012, −3.56306633726354533904744297740, −2.73123077938146838251121187694, −1.55461189943639786045468567311, −0.849039973070957189240485347789, 1.55290188225860965339383496743, 2.67884977988730874826374412182, 3.66733435266661490581066128895, 4.25174743023063751662454771625, 5.10222619559530843675212547993, 6.30714867613046697952901626406, 7.07336821073377879007933534510, 7.969095340289917000045305495235, 8.687518145256356954384629001254, 9.381684380466953954773262847091

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