Properties

Label 2-2888-152.83-c0-0-3
Degree $2$
Conductor $2888$
Sign $0.0238 - 0.999i$
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  + (−0.5 − 0.866i)2-s + (0.939 + 1.62i)3-s + (−0.499 + 0.866i)4-s + (0.939 − 1.62i)6-s + 0.999·8-s + (−1.26 + 2.19i)9-s + 0.347·11-s − 1.87·12-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + 2.53·18-s + (−0.173 − 0.300i)22-s + (0.939 + 1.62i)24-s + (−0.5 + 0.866i)25-s − 2.87·27-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.939 + 1.62i)3-s + (−0.499 + 0.866i)4-s + (0.939 − 1.62i)6-s + 0.999·8-s + (−1.26 + 2.19i)9-s + 0.347·11-s − 1.87·12-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + 2.53·18-s + (−0.173 − 0.300i)22-s + (0.939 + 1.62i)24-s + (−0.5 + 0.866i)25-s − 2.87·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0238 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0238 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.0238 - 0.999i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2819, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.0238 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.182315215\)
\(L(\frac12)\) \(\approx\) \(1.182315215\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 \)
good3 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - 0.347T + T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - 1.53T + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.181104827217375139045018642953, −8.734244492170545490508121988999, −8.033848983258140730640090005897, −7.30826158546439051407038919307, −5.78422545339988296104524075136, −4.90559113488590231640626798433, −4.03723646811873972787289549516, −3.59732534699859407151847611648, −2.76738339141353640070273612832, −1.74901778776636730931783788250, 0.810469590793909577491085084232, 1.82621902203459743614142187765, 2.80825583896444296002341311009, 3.95307650904102647586440589977, 5.20813550873666993337407558325, 6.17042089850362100298032707037, 6.65567159223329309352848942021, 7.41960632250225210865092661947, 7.87434131349372631516717230213, 8.541883933848905682685522657645

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