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Properties

Label 2-2888-152.123-c0-0-3
Degree $2$
Conductor $2888$
Sign $0.714 + 0.700i$
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 + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)6-s + (0.5 + 0.866i)8-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s + (0.766 − 0.642i)16-s + (0.347 + 1.96i)17-s + (0.766 − 0.642i)22-s + (0.939 + 0.342i)24-s + (0.766 + 0.642i)25-s + (0.499 + 0.866i)27-s + (−0.766 − 0.642i)32-s + (0.939 + 0.342i)33-s + (1.87 − 0.684i)34-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)6-s + (0.5 + 0.866i)8-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s + (0.766 − 0.642i)16-s + (0.347 + 1.96i)17-s + (0.766 − 0.642i)22-s + (0.939 + 0.342i)24-s + (0.766 + 0.642i)25-s + (0.499 + 0.866i)27-s + (−0.766 − 0.642i)32-s + (0.939 + 0.342i)33-s + (1.87 − 0.684i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.714 + 0.700i$
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: \((2,\ 2888,\ (\ :0),\ 0.714 + 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.351548020\)
\(L(\frac12)\) \(\approx\) \(1.351548020\)
\(L(1)\) 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 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
5 \( 1 + (-0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (-0.347 - 1.96i)T + (-0.939 + 0.342i)T^{2} \)
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 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (1.87 + 0.684i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (1.53 + 1.28i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
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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

−8.693969863280707446424814144264, −8.404576939099660407216881528732, −7.54119766300800202011269890721, −6.85804750724784958760856829780, −5.69589936947344784667275560320, −4.72635166502805319303727712921, −3.83063502472055764933929353401, −3.09160034479100497144954793305, −1.96394389029809764817339897498, −1.51130941652604428453138521343, 0.923645274946067624005142048956, 2.82641898652551433707878187510, 3.50341200695774901342190382091, 4.50151199209770002992900826768, 5.11028872863095017786895798927, 6.15434123755423510434533787128, 6.76274923356974620945095950224, 7.64878694315030921847091771969, 8.414299334865891283984789439087, 8.915282386766325044860552216063

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