Properties

Label 2-2888-152.123-c0-0-0
Degree $2$
Conductor $2888$
Sign $-0.854 + 0.519i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.854 + 0.519i$
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: \((2,\ 2888,\ (\ :0),\ -0.854 + 0.519i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7269444608\)
\(L(\frac12)\) \(\approx\) \(0.7269444608\)
\(L(1)\) 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

−9.360824406946524616588809363687, −8.465931071345011375132992736094, −7.926970063342483149334594278206, −6.87967449818450164875526747622, −6.38036907238376135502301982330, −5.48899829599636189105423616546, −4.93481174757101640928586508179, −4.14757706686331138797609810281, −3.42724340133725705905199335827, −1.67243791021429230629917055859, 0.51616929634515816952066625186, 1.46458321187513046661434171785, 2.82071232895791343736159884846, 3.46728412778949314313551068734, 4.68709766483578484052480593346, 5.30486207217264703921569866028, 6.16807559125946579169488967316, 6.81537809074580631905196000088, 7.78032448048648459742306757115, 8.757902657083771298384394926780

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