Properties

Label 2-2880-40.29-c1-0-14
Degree $2$
Conductor $2880$
Sign $0.450 - 0.892i$
Analytic cond. $22.9969$
Root an. cond. $4.79550$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.18 − 0.456i)5-s + 0.913i·7-s − 3.58i·11-s + 0.913·13-s + 3.58i·17-s + 4i·19-s + (4.58 + 1.99i)25-s − 7.84i·29-s − 5.29·31-s + (0.417 − 1.99i)35-s − 7.84·37-s − 6·41-s + 7.16·43-s + 6.92i·47-s + 6.16·49-s + ⋯
L(s)  = 1  + (−0.978 − 0.204i)5-s + 0.345i·7-s − 1.08i·11-s + 0.253·13-s + 0.868i·17-s + 0.917i·19-s + (0.916 + 0.399i)25-s − 1.45i·29-s − 0.950·31-s + (0.0705 − 0.338i)35-s − 1.28·37-s − 0.937·41-s + 1.09·43-s + 1.01i·47-s + 0.880·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.450 - 0.892i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.450 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.092791199\)
\(L(\frac12)\) \(\approx\) \(1.092791199\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (2.18 + 0.456i)T \)
good7 \( 1 - 0.913iT - 7T^{2} \)
11 \( 1 + 3.58iT - 11T^{2} \)
13 \( 1 - 0.913T + 13T^{2} \)
17 \( 1 - 3.58iT - 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 7.84iT - 29T^{2} \)
31 \( 1 + 5.29T + 31T^{2} \)
37 \( 1 + 7.84T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 7.16T + 43T^{2} \)
47 \( 1 - 6.92iT - 47T^{2} \)
53 \( 1 + 2.55T + 53T^{2} \)
59 \( 1 + 7.58iT - 59T^{2} \)
61 \( 1 - 10.5iT - 61T^{2} \)
67 \( 1 - 15.1T + 67T^{2} \)
71 \( 1 - 6.92T + 71T^{2} \)
73 \( 1 - 12iT - 73T^{2} \)
79 \( 1 - 5.29T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 7.16iT - 97T^{2} \)
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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

−8.701595230978895987997291269830, −8.192410910778425023805689647899, −7.62387153272583307716291957397, −6.56249041352521478958023101113, −5.84913683795682544708805031810, −5.09569694072526889435018610233, −3.84486003475757228115059896243, −3.62278397419270108024858030187, −2.29894247864117417237016901876, −0.957336320446411597133264084955, 0.43584334449993357908251174287, 1.89784122970184429528566597000, 3.08965076835682512648064748049, 3.83911447074284726719858156589, 4.74839102576623973182563187368, 5.31374102775454137755547242767, 6.83142312639007721758214686238, 7.01454309207103059599966613688, 7.73798484233679115927359868705, 8.687076525145626034533847247237

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