Properties

Label 2-2880-5.4-c1-0-36
Degree $2$
Conductor $2880$
Sign $0.447 + 0.894i$
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  + (1 + 2i)5-s − 4i·7-s − 4·11-s + 4i·17-s − 4i·23-s + (−3 + 4i)25-s + 6·29-s + 4·31-s + (8 − 4i)35-s − 8i·37-s + 10·41-s − 4i·43-s − 4i·47-s − 9·49-s − 12i·53-s + ⋯
L(s)  = 1  + (0.447 + 0.894i)5-s − 1.51i·7-s − 1.20·11-s + 0.970i·17-s − 0.834i·23-s + (−0.600 + 0.800i)25-s + 1.11·29-s + 0.718·31-s + (1.35 − 0.676i)35-s − 1.31i·37-s + 1.56·41-s − 0.609i·43-s − 0.583i·47-s − 1.28·49-s − 1.64i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.554183671\)
\(L(\frac12)\) \(\approx\) \(1.554183671\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 2i)T \)
good7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 8iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−8.444704966249952465440110312309, −7.80236829973923599195956978982, −7.11790515786410820470321984898, −6.47634634599677172449226370459, −5.67699219058747295802009781864, −4.63509381443500709508140664893, −3.84865174102725762780628519047, −2.93200345946838934148684090242, −1.99396419588046229985214049995, −0.53434555950747201256705787350, 1.13630905254918138951108820358, 2.46351656050060685852924316645, 2.88183198193585032003604208993, 4.51479370468263479328035015029, 5.09908016536640263023701841801, 5.72798517486197668433087073447, 6.38817983545297678394646740654, 7.66199205490648303859296252261, 8.195899155635972310779879105011, 8.975964025100136206621664613492

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