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Properties

Label 2-2880-120.77-c1-0-26
Degree $2$
Conductor $2880$
Sign $0.733 + 0.679i$
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  + (0.224 + 2.22i)5-s + (−1.41 − 1.41i)7-s − 3.46·11-s + (3.14 + 3.14i)13-s + (4.87 − 4.87i)17-s − 6.89·19-s + (−4.44 − 4.44i)23-s + (−4.89 + i)25-s − 8.44i·29-s + 9.75·31-s + (2.82 − 3.46i)35-s + (2.51 − 2.51i)37-s + 1.41i·41-s + (6.89 + 6.89i)43-s + (−1.55 + 1.55i)47-s + ⋯
L(s)  = 1  + (0.100 + 0.994i)5-s + (−0.534 − 0.534i)7-s − 1.04·11-s + (0.872 + 0.872i)13-s + (1.18 − 1.18i)17-s − 1.58·19-s + (−0.927 − 0.927i)23-s + (−0.979 + 0.200i)25-s − 1.56i·29-s + 1.75·31-s + (0.478 − 0.585i)35-s + (0.412 − 0.412i)37-s + 0.220i·41-s + (1.05 + 1.05i)43-s + (−0.226 + 0.226i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.733 + 0.679i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.733 + 0.679i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.342256271\)
\(L(\frac12)\) \(\approx\) \(1.342256271\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.224 - 2.22i)T \)
good7 \( 1 + (1.41 + 1.41i)T + 7iT^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 + (-3.14 - 3.14i)T + 13iT^{2} \)
17 \( 1 + (-4.87 + 4.87i)T - 17iT^{2} \)
19 \( 1 + 6.89T + 19T^{2} \)
23 \( 1 + (4.44 + 4.44i)T + 23iT^{2} \)
29 \( 1 + 8.44iT - 29T^{2} \)
31 \( 1 - 9.75T + 31T^{2} \)
37 \( 1 + (-2.51 + 2.51i)T - 37iT^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 + (-6.89 - 6.89i)T + 43iT^{2} \)
47 \( 1 + (1.55 - 1.55i)T - 47iT^{2} \)
53 \( 1 + (-6.89 + 6.89i)T - 53iT^{2} \)
59 \( 1 - 0.635iT - 59T^{2} \)
61 \( 1 - 7.56iT - 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 3.10iT - 71T^{2} \)
73 \( 1 + (-1.89 + 1.89i)T - 73iT^{2} \)
79 \( 1 + 14.1iT - 79T^{2} \)
83 \( 1 + (2.19 - 2.19i)T - 83iT^{2} \)
89 \( 1 - 4.24T + 89T^{2} \)
97 \( 1 + (-1.89 - 1.89i)T + 97iT^{2} \)
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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

−8.536310075168256042706187097245, −7.88114188811078503656503873217, −7.15598604169867086694364732631, −6.27455107326516695535441708969, −6.00184356440534513464681637368, −4.58923161952666701835115428656, −3.93341744862305257325194510952, −2.87351333044134921981623694895, −2.23126258415797410009733333440, −0.50207241551992249699250593473, 1.00043511445656233279100510795, 2.16511268653971724015487298957, 3.27225887931057581566408156498, 4.09599861748475705247985242166, 5.15704141186976218531716874483, 5.82020297933499455240096463308, 6.25587729639931292977076337790, 7.62280180763347758978678648990, 8.303339576335824797715390286704, 8.610014285572772485158700701490

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