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Properties

Label 2-2888-152.11-c0-0-6
Degree $2$
Conductor $2888$
Sign $-0.877 - 0.479i$
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  + (0.5 − 0.866i)2-s + (0.766 − 1.32i)3-s + (−0.499 − 0.866i)4-s + (−0.766 − 1.32i)6-s − 0.999·8-s + (−0.673 − 1.16i)9-s − 1.87·11-s − 1.53·12-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s − 1.34·18-s + (−0.939 + 1.62i)22-s + (−0.766 + 1.32i)24-s + (−0.5 − 0.866i)25-s − 0.532·27-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.766 − 1.32i)3-s + (−0.499 − 0.866i)4-s + (−0.766 − 1.32i)6-s − 0.999·8-s + (−0.673 − 1.16i)9-s − 1.87·11-s − 1.53·12-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s − 1.34·18-s + (−0.939 + 1.62i)22-s + (−0.766 + 1.32i)24-s + (−0.5 − 0.866i)25-s − 0.532·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.877 - 0.479i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2595, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ -0.877 - 0.479i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.494129439\)
\(L(\frac12)\) \(\approx\) \(1.494129439\)
\(L(1)\) 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 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + 1.87T + T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - 0.347T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.490925130594945689325255863470, −7.71875836608039642900441569258, −7.22242429676782956632471905593, −6.10895293720135476211801084025, −5.42540315330864208121134953011, −4.52994645473897548475137041408, −3.28965480465867525004604344018, −2.59667748254461379276384697574, −2.04233655276929391648399907276, −0.68691444848342487542944330487, 2.44279627827737055530011346542, 3.24984071339932154223625044144, 3.91991068453091933494246868869, 4.81799109616665300383974599061, 5.35019369014181707417085910912, 6.11914765123222588295676330592, 7.35230470637835770301885784296, 8.005528855760030959865925186140, 8.423846854025938528746810104625, 9.348777737947933061062941086317

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