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Properties

Label 2-2880-60.59-c1-0-3
Degree $2$
Conductor $2880$
Sign $0.0691 - 0.997i$
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  + (−1.73 − 1.41i)5-s − 4.24·7-s + 2.44·11-s − 2.44i·13-s − 6.92·17-s − 6.92i·19-s − 6i·23-s + (0.999 + 4.89i)25-s + 2.82i·29-s + 3.46i·31-s + (7.34 + 6i)35-s + 2.44i·37-s + 7.07i·41-s + 8.48·43-s + 10.9·49-s + ⋯
L(s)  = 1  + (−0.774 − 0.632i)5-s − 1.60·7-s + 0.738·11-s − 0.679i·13-s − 1.68·17-s − 1.58i·19-s − 1.25i·23-s + (0.199 + 0.979i)25-s + 0.525i·29-s + 0.622i·31-s + (1.24 + 1.01i)35-s + 0.402i·37-s + 1.10i·41-s + 1.29·43-s + 1.57·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0691 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0691 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.0691 - 0.997i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2879, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.0691 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3280150126\)
\(L(\frac12)\) \(\approx\) \(0.3280150126\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 1.41i)T \)
good7 \( 1 + 4.24T + 7T^{2} \)
11 \( 1 - 2.44T + 11T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 + 6.92T + 17T^{2} \)
19 \( 1 + 6.92iT - 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 - 2.44iT - 37T^{2} \)
41 \( 1 - 7.07iT - 41T^{2} \)
43 \( 1 - 8.48T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 9.79T + 71T^{2} \)
73 \( 1 - 4.89iT - 73T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 7.07iT - 89T^{2} \)
97 \( 1 - 14.6iT - 97T^{2} \)
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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

−9.045228792075484802369571431075, −8.407067815013716767606803494855, −7.27629774102644592962284054016, −6.69175194952005342572092950287, −6.12691658388867627330133999583, −4.84515295956167974720901929629, −4.30762413478364999844440379165, −3.30896075098848865546844955685, −2.58022570843821904992918452196, −0.842608980815016461705961192752, 0.13822761797608785625732089284, 1.93658894330022312931645099829, 3.04172708649602244484488712535, 3.87919108407360880243743370804, 4.24908991247745574617744006877, 5.88841037520480075100988116790, 6.31515870886820138687842554062, 7.07062374642523844786114539216, 7.63122687848101515195583127550, 8.749195219299279715930586438470

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