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Properties

Label 2-2888-152.83-c0-0-0
Degree $2$
Conductor $2888$
Sign $0.928 + 0.370i$
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 + (−0.766 − 1.32i)3-s + (−0.499 + 0.866i)4-s + (−0.766 + 1.32i)6-s + 0.999·8-s + (−0.673 + 1.16i)9-s − 1.87·11-s + 1.53·12-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + 1.34·18-s + (0.939 + 1.62i)22-s + (−0.766 − 1.32i)24-s + (−0.5 + 0.866i)25-s + 0.532·27-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.766 − 1.32i)3-s + (−0.499 + 0.866i)4-s + (−0.766 + 1.32i)6-s + 0.999·8-s + (−0.673 + 1.16i)9-s − 1.87·11-s + 1.53·12-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + 1.34·18-s + (0.939 + 1.62i)22-s + (−0.766 − 1.32i)24-s + (−0.5 + 0.866i)25-s + 0.532·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.928 + 0.370i$
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: \((2,\ 2888,\ (\ :0),\ 0.928 + 0.370i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3807894399\)
\(L(\frac12)\) \(\approx\) \(0.3807894399\)
\(L(1)\) 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 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + 1.87T + T^{2} \)
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 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - 0.347T + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.790502181771245745105051728810, −7.942450421150700281042733495019, −7.64334318047952149915461409435, −6.87401846674390821780177885824, −5.72978629874290861574503459941, −5.27368729736430391520583308511, −4.03402513160551732087726176585, −2.85177531042252682158178136985, −2.07592113183844305123237163760, −1.05357064850437119726910331811, 0.36831052152737027040768285476, 2.41194007802340266673115447423, 3.71863990237356771452906315787, 4.71816454638715442129940559056, 5.22238252801421634228110642134, 5.71160676567331818298568769050, 6.64344738235392602283199797012, 7.65393448501170312365360368818, 8.134508895292597931747914002798, 9.159541622255110431546318405928

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