Properties

Label 2-2888-152.35-c0-0-6
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  + (0.766 + 0.642i)2-s + (1.76 − 0.642i)3-s + (0.173 + 0.984i)4-s + (1.76 + 0.642i)6-s + (−0.500 + 0.866i)8-s + (1.93 − 1.62i)9-s + (−0.173 + 0.300i)11-s + (0.939 + 1.62i)12-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s + 2.53·18-s + (−0.326 + 0.118i)22-s + (−0.326 + 1.85i)24-s + (−0.939 − 0.342i)25-s + (1.43 − 2.49i)27-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (1.76 − 0.642i)3-s + (0.173 + 0.984i)4-s + (1.76 + 0.642i)6-s + (−0.500 + 0.866i)8-s + (1.93 − 1.62i)9-s + (−0.173 + 0.300i)11-s + (0.939 + 1.62i)12-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s + 2.53·18-s + (−0.326 + 0.118i)22-s + (−0.326 + 1.85i)24-s + (−0.939 − 0.342i)25-s + (1.43 − 2.49i)27-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: $\chi_{2888} (1859, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.854 - 0.519i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.238179342\)
\(L(\frac12)\) \(\approx\) \(3.238179342\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 - 0.642i)T \)
19 \( 1 \)
good3 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
5 \( 1 + (0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
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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

−8.817752783677320103136559892878, −7.985611599063441443597862973735, −7.67118580597858557813780245629, −6.80611959826349112229054261516, −6.30335289047551213423021650379, −4.97747714664260495981443775951, −4.19615159672880746910961112043, −3.35999494955568679316791089138, −2.62594011442211910364041881227, −1.82203085985544451982701979060, 1.77356914607159640959250857582, 2.38864530148281236270064334462, 3.42729402701462546851841921927, 3.82131075727552852412857379963, 4.67870156052823582254878464448, 5.50050163419007950546888852269, 6.65039080449053016312132158146, 7.46956369552738232636715444644, 8.488062942817625292891750516113, 8.818534808166803290555612683672

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