Properties

Label 2-2888-152.99-c0-0-3
Degree $2$
Conductor $2888$
Sign $0.974 + 0.226i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.939 + 0.342i)2-s + (−0.266 − 1.50i)3-s + (0.766 + 0.642i)4-s + (0.266 − 1.50i)6-s + (0.500 + 0.866i)8-s + (−1.26 + 0.460i)9-s + (0.939 + 1.62i)11-s + (0.766 − 1.32i)12-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s − 1.34·18-s + (0.326 + 1.85i)22-s + (1.17 − 0.984i)24-s + (0.173 − 0.984i)25-s + (0.266 + 0.460i)27-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (−0.266 − 1.50i)3-s + (0.766 + 0.642i)4-s + (0.266 − 1.50i)6-s + (0.500 + 0.866i)8-s + (−1.26 + 0.460i)9-s + (0.939 + 1.62i)11-s + (0.766 − 1.32i)12-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s − 1.34·18-s + (0.326 + 1.85i)22-s + (1.17 − 0.984i)24-s + (0.173 − 0.984i)25-s + (0.266 + 0.460i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.974 + 0.226i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.974 + 0.226i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.176136323\)
\(L(\frac12)\) \(\approx\) \(2.176136323\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 \)
good3 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
5 \( 1 + (-0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + (-0.173 - 0.984i)T^{2} \)
59 \( 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.544679475265943740565289122925, −7.73717709063548751242154013417, −7.32047573159941150570473799155, −6.49889751258171344595016022084, −6.21964829309436929228184720959, −5.11723389957821482909291379196, −4.36280300032940901537312320996, −3.29945266027797467412227052557, −2.12936087948683535619722696861, −1.50046052541945540019839182565, 1.25934674060609043023673179723, 3.01477067710761075055830759071, 3.46813126143027143253076556522, 4.18113332209920085117323736000, 5.05687833599421377032916341931, 5.67200295817595962707949553802, 6.26503201638558077053712507202, 7.30048987790797324844280462833, 8.446508463984617579685226420452, 9.272271793122195410210920798728

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