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Properties

Label 2-2888-2888.2531-c0-0-0
Degree $2$
Conductor $2888$
Sign $0.969 - 0.243i$
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  + (−0.999 − 0.0183i)2-s + (1.14 − 0.766i)3-s + (0.999 + 0.0367i)4-s + (−1.16 + 0.744i)6-s + (−0.998 − 0.0550i)8-s + (0.349 − 0.838i)9-s + (−0.442 + 1.41i)11-s + (1.17 − 0.723i)12-s + (0.997 + 0.0734i)16-s + (1.02 + 1.63i)17-s + (−0.364 + 0.831i)18-s + (0.515 + 0.856i)19-s + (0.468 − 1.40i)22-s + (−1.18 + 0.701i)24-s + (−0.621 + 0.783i)25-s + ⋯
L(s)  = 1  + (−0.999 − 0.0183i)2-s + (1.14 − 0.766i)3-s + (0.999 + 0.0367i)4-s + (−1.16 + 0.744i)6-s + (−0.998 − 0.0550i)8-s + (0.349 − 0.838i)9-s + (−0.442 + 1.41i)11-s + (1.17 − 0.723i)12-s + (0.997 + 0.0734i)16-s + (1.02 + 1.63i)17-s + (−0.364 + 0.831i)18-s + (0.515 + 0.856i)19-s + (0.468 − 1.40i)22-s + (−1.18 + 0.701i)24-s + (−0.621 + 0.783i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.969 - 0.243i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.969 - 0.243i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.158869333\)
\(L(\frac12)\) \(\approx\) \(1.158869333\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.999 + 0.0183i)T \)
19 \( 1 + (-0.515 - 0.856i)T \)
good3 \( 1 + (-1.14 + 0.766i)T + (0.384 - 0.922i)T^{2} \)
5 \( 1 + (0.621 - 0.783i)T^{2} \)
7 \( 1 + (-0.716 + 0.697i)T^{2} \)
11 \( 1 + (0.442 - 1.41i)T + (-0.821 - 0.569i)T^{2} \)
13 \( 1 + (-0.888 + 0.459i)T^{2} \)
17 \( 1 + (-1.02 - 1.63i)T + (-0.435 + 0.900i)T^{2} \)
23 \( 1 + (-0.606 + 0.794i)T^{2} \)
29 \( 1 + (-0.811 - 0.584i)T^{2} \)
31 \( 1 + (-0.137 + 0.990i)T^{2} \)
37 \( 1 + (0.0825 - 0.996i)T^{2} \)
41 \( 1 + (0.187 + 0.529i)T + (-0.777 + 0.628i)T^{2} \)
43 \( 1 + (0.221 + 0.313i)T + (-0.333 + 0.942i)T^{2} \)
47 \( 1 + (-0.577 - 0.816i)T^{2} \)
53 \( 1 + (0.995 - 0.0917i)T^{2} \)
59 \( 1 + (0.305 + 0.0568i)T + (0.933 + 0.359i)T^{2} \)
61 \( 1 + (0.971 + 0.236i)T^{2} \)
67 \( 1 + (-0.451 + 1.93i)T + (-0.896 - 0.443i)T^{2} \)
71 \( 1 + (0.971 - 0.236i)T^{2} \)
73 \( 1 + (-0.311 + 0.589i)T + (-0.562 - 0.826i)T^{2} \)
79 \( 1 + (-0.983 + 0.182i)T^{2} \)
83 \( 1 + (1.56 + 0.441i)T + (0.851 + 0.523i)T^{2} \)
89 \( 1 + (0.670 + 1.26i)T + (-0.562 + 0.826i)T^{2} \)
97 \( 1 + (0.348 + 1.49i)T + (-0.896 + 0.443i)T^{2} \)
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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

−8.781672035161134362538285053415, −8.199651720702446241649004355473, −7.54250669528147132048167897842, −7.27077086720708605467920542895, −6.21179441469918810417752546036, −5.35646119116057822619843910601, −3.84374896289861277629279101015, −3.09993828360143003015474096078, −1.92980849042090190411251350753, −1.61209058789696425153855917336, 0.905326711220570366679568898044, 2.69762456275014006736205405567, 2.87021248324133898659955362103, 3.86741180391861126725439543715, 5.12480508249219580897836787869, 5.90383489068764834078818963672, 6.97742385714238294361757633836, 7.76011354483602015569574947091, 8.331778135289259295972002387429, 8.909397346592111331335179405704

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