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Properties

Label 2-2888-152.13-c0-0-1
Degree $2$
Conductor $2888$
Sign $0.158 + 0.987i$
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  + (0.766 − 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (0.939 − 0.342i)6-s + (0.5 − 0.866i)7-s + (−0.500 − 0.866i)8-s + (0.499 − 0.866i)12-s + (0.939 − 0.342i)13-s + (−0.173 − 0.984i)14-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s + (0.766 − 0.642i)21-s + (−0.173 + 0.984i)23-s + (−0.173 − 0.984i)24-s + (−0.939 + 0.342i)25-s + (0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (0.939 − 0.342i)6-s + (0.5 − 0.866i)7-s + (−0.500 − 0.866i)8-s + (0.499 − 0.866i)12-s + (0.939 − 0.342i)13-s + (−0.173 − 0.984i)14-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s + (0.766 − 0.642i)21-s + (−0.173 + 0.984i)23-s + (−0.173 − 0.984i)24-s + (−0.939 + 0.342i)25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.158 + 0.987i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.158 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.570937469\)
\(L(\frac12)\) \(\approx\) \(2.570937469\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 \)
good3 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
5 \( 1 + (0.939 - 0.342i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
29 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - 2T + T^{2} \)
41 \( 1 + (-0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.939 - 0.342i)T^{2} \)
47 \( 1 + (-1.53 - 1.28i)T + (0.173 + 0.984i)T^{2} \)
53 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
59 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.766 + 0.642i)T^{2} \)
97 \( 1 + (-0.173 + 0.984i)T^{2} \)
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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

−9.039595094799104568286263309951, −7.973050191379790988848004046344, −7.48132727229362027685125182894, −6.14714849306925550611210927578, −5.77761721116739746037374203978, −4.34630084337442517755839772711, −4.07017028133501464171808483539, −3.26466141831699413286017230888, −2.29879396054893232067188507500, −1.23975014860829332595279265172, 2.04425037369161743377614303855, 2.54569677714035643723894144212, 3.59642277608225738382707716345, 4.41879634810256802453526232497, 5.34632040136139202675750114620, 6.05470782512826033087256583503, 6.84175561773707240901384118070, 7.71214043812003690708958519178, 8.278669968715083776459683061624, 8.848601459222200494409246312141

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