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Properties

Label 2-2880-120.53-c1-0-37
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: $\chi_{2880} (1313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.733 + 0.679i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1711386855\)
\(L(\frac12)\) \(\approx\) \(0.1711386855\)
\(L(\frac{3}{2})\) 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

−8.300983625347371784346721715707, −7.64946133323100223086740706003, −7.11431995615296935425438181720, −6.25657665822618075618025682415, −5.46051750202003878939279035236, −4.40173818525544238855119775210, −3.79704779922928562012018594352, −2.58972881302136711199648659915, −1.90134298041377961753716534762, −0.05239333966105171570234510855, 1.32967801867275666417526736575, 2.48306486893785869636420173832, 3.37045635318452175102819753242, 4.66590414697626908578726297847, 5.35512703595491676759382614518, 5.47760823920211711413844145596, 6.98624135899040717597750220489, 7.76853450345293056929394998340, 8.167784394526363464757552027858, 9.135053452439784254782899163332

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