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Properties

Label 2-2880-48.11-c1-0-10
Degree $2$
Conductor $2880$
Sign $0.509 - 0.860i$
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  + (0.707 − 0.707i)5-s + 0.750·7-s + (2.08 + 2.08i)11-s + (−1.66 + 1.66i)13-s − 1.81i·17-s + (1.22 + 1.22i)19-s + 9.36i·23-s − 1.00i·25-s + (−4.84 − 4.84i)29-s + 4.95i·31-s + (0.530 − 0.530i)35-s + (5.94 + 5.94i)37-s + 2.47·41-s + (−5.20 + 5.20i)43-s + 4.43·47-s + ⋯
L(s)  = 1  + (0.316 − 0.316i)5-s + 0.283·7-s + (0.628 + 0.628i)11-s + (−0.461 + 0.461i)13-s − 0.439i·17-s + (0.281 + 0.281i)19-s + 1.95i·23-s − 0.200i·25-s + (−0.900 − 0.900i)29-s + 0.889i·31-s + (0.0896 − 0.0896i)35-s + (0.976 + 0.976i)37-s + 0.386·41-s + (−0.793 + 0.793i)43-s + 0.647·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.509 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.509 - 0.860i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1871, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.509 - 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.833291606\)
\(L(\frac12)\) \(\approx\) \(1.833291606\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 - 0.750T + 7T^{2} \)
11 \( 1 + (-2.08 - 2.08i)T + 11iT^{2} \)
13 \( 1 + (1.66 - 1.66i)T - 13iT^{2} \)
17 \( 1 + 1.81iT - 17T^{2} \)
19 \( 1 + (-1.22 - 1.22i)T + 19iT^{2} \)
23 \( 1 - 9.36iT - 23T^{2} \)
29 \( 1 + (4.84 + 4.84i)T + 29iT^{2} \)
31 \( 1 - 4.95iT - 31T^{2} \)
37 \( 1 + (-5.94 - 5.94i)T + 37iT^{2} \)
41 \( 1 - 2.47T + 41T^{2} \)
43 \( 1 + (5.20 - 5.20i)T - 43iT^{2} \)
47 \( 1 - 4.43T + 47T^{2} \)
53 \( 1 + (-7.54 + 7.54i)T - 53iT^{2} \)
59 \( 1 + (-0.472 - 0.472i)T + 59iT^{2} \)
61 \( 1 + (4.99 - 4.99i)T - 61iT^{2} \)
67 \( 1 + (1.95 + 1.95i)T + 67iT^{2} \)
71 \( 1 - 0.418iT - 71T^{2} \)
73 \( 1 - 3.49iT - 73T^{2} \)
79 \( 1 - 3.40iT - 79T^{2} \)
83 \( 1 + (0.548 - 0.548i)T - 83iT^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 - 6.64T + 97T^{2} \)
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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

−9.095500131706512405096778449267, −8.053022327415328782762324939481, −7.42726960993590865913030922720, −6.68076691601466475857755256490, −5.76535514599578354453873400538, −5.01673896423740419014763096429, −4.26339130821768948197502534444, −3.30598486675119419058095548740, −2.08495647445089842044139302420, −1.25966429054220825057487100634, 0.61603775472053574067389958078, 1.98260302516501327502255725107, 2.89148901707414333112273970421, 3.87387249208678637732602122396, 4.73892219902127867944041841506, 5.70441016363060559745104059653, 6.28976125384493498389259926328, 7.15251666025974966566814623625, 7.87331577816656862904979439336, 8.743907698301360233608895471406

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