Properties

Label 2-2888-152.99-c0-0-5
Degree $2$
Conductor $2888$
Sign $-0.600 + 0.799i$
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 + (−0.500 − 0.866i)8-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)12-s + (0.173 + 0.984i)16-s + (−1.87 − 0.684i)17-s + (−0.173 − 0.984i)22-s + (−0.766 + 0.642i)24-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)27-s + (0.173 − 0.984i)32-s + (0.766 − 0.642i)33-s + (1.53 + 1.28i)34-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 + (−0.500 − 0.866i)8-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)12-s + (0.173 + 0.984i)16-s + (−1.87 − 0.684i)17-s + (−0.173 − 0.984i)22-s + (−0.766 + 0.642i)24-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)27-s + (0.173 − 0.984i)32-s + (0.766 − 0.642i)33-s + (1.53 + 1.28i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.600 + 0.799i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ -0.600 + 0.799i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6649874703\)
\(L(\frac12)\) \(\approx\) \(0.6649874703\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (1.87 + 0.684i)T + (0.766 + 0.642i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-1.53 + 1.28i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + (-0.173 - 0.984i)T^{2} \)
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 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.347 + 1.96i)T + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
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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

−8.804620283198922509362667604369, −7.891914451069816741927300565796, −7.06092673917340485063895094668, −6.85078991123578969011660908325, −6.04900231678502745651434416881, −4.65551833989971582584397563549, −3.82228519154998574299724434764, −2.36719589324225337726965757802, −1.95828242719723953100016845398, −0.60910059503852922791290117459, 1.34219073662482448052155765958, 2.58776253955292085717042848389, 3.75739115220094840020928697475, 4.58378914598783496287340410022, 5.48519093779802920444481530902, 6.30783012334885212419549940503, 6.90267644255066307415390916447, 7.912300848932111071654926253697, 8.625721937804754221439754983364, 9.282853918995797156868021930839

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