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Properties

Label 2-2888-152.21-c0-0-1
Degree $2$
Conductor $2888$
Sign $-0.486 - 0.873i$
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  + (−0.173 + 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (−0.766 + 0.642i)6-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.499 − 0.866i)12-s + (0.766 − 0.642i)13-s + (−0.939 + 0.342i)14-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s + (−0.173 + 0.984i)21-s + (0.939 + 0.342i)23-s + (0.939 − 0.342i)24-s + (0.766 − 0.642i)25-s + (0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (−0.766 + 0.642i)6-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.499 − 0.866i)12-s + (0.766 − 0.642i)13-s + (−0.939 + 0.342i)14-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s + (−0.173 + 0.984i)21-s + (0.939 + 0.342i)23-s + (0.939 − 0.342i)24-s + (0.766 − 0.642i)25-s + (0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.486 - 0.873i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (477, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ -0.486 - 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.464353218\)
\(L(\frac12)\) \(\approx\) \(1.464353218\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 \)
good3 \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \)
5 \( 1 + (-0.766 + 0.642i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 2T + T^{2} \)
41 \( 1 + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (-0.347 - 1.96i)T + (-0.939 + 0.342i)T^{2} \)
53 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.173 + 0.984i)T^{2} \)
97 \( 1 + (0.939 + 0.342i)T^{2} \)
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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

−8.827693263426732243548797903977, −8.599408087440597694476867418251, −7.989347371110957873463749916554, −6.91957598483210618002447179928, −6.17090498530175976737027423460, −5.37636818290611009874821690400, −4.67363137297238249845688515887, −3.70344844579570696876026231640, −2.96246869729841012150783096854, −1.44297436389906666318456100720, 1.07024134356112830537930207254, 1.92355779555376609970812347106, 2.87267701004437449148950403320, 3.69305731388831612042068609434, 4.60737062821258431640247222085, 5.31555142184851558962619040473, 6.83663449704211842075884705688, 7.30987085699735184612785784076, 8.137465274611785809562582949402, 8.790313952760150193080725258882

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