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Properties

Label 2-2888-152.123-c0-0-4
Degree $2$
Conductor $2888$
Sign $-0.320 + 0.947i$
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 + (−0.266 + 0.223i)3-s + (−0.939 + 0.342i)4-s + (0.266 + 0.223i)6-s + (0.5 + 0.866i)8-s + (−0.152 + 0.866i)9-s + (−0.766 − 1.32i)11-s + (0.173 − 0.300i)12-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + 0.879·18-s + (−1.17 + 0.984i)22-s + (−0.326 − 0.118i)24-s + (0.766 + 0.642i)25-s + (−0.326 − 0.565i)27-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.266 + 0.223i)3-s + (−0.939 + 0.342i)4-s + (0.266 + 0.223i)6-s + (0.5 + 0.866i)8-s + (−0.152 + 0.866i)9-s + (−0.766 − 1.32i)11-s + (0.173 − 0.300i)12-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + 0.879·18-s + (−1.17 + 0.984i)22-s + (−0.326 − 0.118i)24-s + (0.766 + 0.642i)25-s + (−0.326 − 0.565i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.320 + 0.947i$
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: \((2,\ 2888,\ (\ :0),\ -0.320 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7548940463\)
\(L(\frac12)\) \(\approx\) \(0.7548940463\)
\(L(1)\) 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 \( 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2} \)
5 \( 1 + (-0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
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 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \)
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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

−8.968237667472593964308513722936, −8.002205171608649250853049570174, −7.61637237882617392739158044035, −6.28694151884376407231523547033, −5.21544600684911770182465188639, −4.99480829377312303487336542660, −3.77082388704072490170544603374, −2.93725894377190229711552888042, −2.15111313930195913063104150678, −0.60647879965516300751839247632, 1.17052212995568422422147514703, 2.61097146011034204308030634218, 3.98539576257835080118937488202, 4.59695926190580993289383868058, 5.55828918769739206273958758874, 6.22540698324612100592540337893, 6.91218848227386259220066398607, 7.57319669663218234230541428815, 8.296315663433176038364362448114, 9.085700874454810343596894067043

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