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Properties

Label 2-2888-152.109-c0-0-1
Degree $2$
Conductor $2888$
Sign $0.378 + 0.925i$
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 + (0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.173 − 0.984i)6-s + (0.5 + 0.866i)7-s + (0.500 − 0.866i)8-s + (−0.5 − 0.866i)12-s + (0.173 + 0.984i)13-s + (0.766 + 0.642i)14-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + (0.939 − 0.342i)21-s + (−0.766 + 0.642i)23-s + (−0.766 − 0.642i)24-s + (0.173 + 0.984i)25-s + (0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.173 − 0.984i)6-s + (0.5 + 0.866i)7-s + (0.500 − 0.866i)8-s + (−0.5 − 0.866i)12-s + (0.173 + 0.984i)13-s + (0.766 + 0.642i)14-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + (0.939 − 0.342i)21-s + (−0.766 + 0.642i)23-s + (−0.766 − 0.642i)24-s + (0.173 + 0.984i)25-s + (0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.378 + 0.925i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.378 + 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.592226248\)
\(L(\frac12)\) \(\approx\) \(2.592226248\)
\(L(1)\) 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 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
5 \( 1 + (-0.173 - 0.984i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 2T + T^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.173 - 0.984i)T^{2} \)
47 \( 1 + (1.87 + 0.684i)T + (0.766 + 0.642i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
59 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.173 + 0.984i)T^{2} \)
67 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.939 - 0.342i)T^{2} \)
97 \( 1 + (-0.766 + 0.642i)T^{2} \)
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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

−8.744363192784930897321500821018, −7.82204998226792337518212212441, −7.18615924972471862552597952745, −6.52774437326228030241990723077, −5.60388848093231338754180399671, −5.09450502804641131273600086334, −3.98551967198951257638640478051, −3.10786230711116819668335042260, −1.89787353921309520428918247805, −1.62497127028816034218825278292, 1.59592751060580728261813543724, 3.06206898268024071980378589579, 3.68248748438174691994156191238, 4.39969466412613144149664653094, 5.05149506919231080578216474707, 5.84442100472926051764147726702, 6.74546967741827125935311857998, 7.62659828793603534942575678213, 8.123606209540318977044892358585, 8.995862438182720560129959864959

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