Properties

Label 2-2888-152.35-c0-0-0
Degree $2$
Conductor $2888$
Sign $-0.363 - 0.931i$
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 + (0.326 − 0.118i)3-s + (0.173 + 0.984i)4-s + (−0.326 − 0.118i)6-s + (0.500 − 0.866i)8-s + (−0.673 + 0.565i)9-s + (−0.766 + 1.32i)11-s + (0.173 + 0.300i)12-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s + 0.879·18-s + (1.43 − 0.524i)22-s + (0.0603 − 0.342i)24-s + (−0.939 − 0.342i)25-s + (−0.326 + 0.565i)27-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.326 − 0.118i)3-s + (0.173 + 0.984i)4-s + (−0.326 − 0.118i)6-s + (0.500 − 0.866i)8-s + (−0.673 + 0.565i)9-s + (−0.766 + 1.32i)11-s + (0.173 + 0.300i)12-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s + 0.879·18-s + (1.43 − 0.524i)22-s + (0.0603 − 0.342i)24-s + (−0.939 − 0.342i)25-s + (−0.326 + 0.565i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.363 - 0.931i$
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: \((2,\ 2888,\ (\ :0),\ -0.363 - 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3264967351\)
\(L(\frac12)\) \(\approx\) \(0.3264967351\)
\(L(1)\) 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 \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \)
5 \( 1 + (0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \)
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 \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (-1.43 - 1.20i)T + (0.173 + 0.984i)T^{2} \)
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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

−9.378623351359094034659401414613, −8.313668722892315862808242573043, −7.949461419968160632042873099519, −7.20548987273692779338045741083, −6.41212603316914391826751411886, −5.09471918037778321953593550912, −4.46389024739387989570389346269, −3.25559517346108979056305973019, −2.44304109337064591319266065701, −1.77459382425977294508705141417, 0.23125042384434741832610571197, 1.85088629190448296435541598871, 2.95986115928077477386995662327, 3.89263359034411314783900896050, 5.17195224370701747702947842696, 5.81895619500669336976867520901, 6.42949714246979090534347078982, 7.33339325268218174547959317123, 8.244441776892953788713582975148, 8.586710958407237614858675650121

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