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Properties

Label 2-2880-48.35-c1-0-11
Degree $2$
Conductor $2880$
Sign $0.991 - 0.127i$
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 − 3.83·7-s + (0.214 − 0.214i)11-s + (−0.177 − 0.177i)13-s − 1.29i·17-s + (−2.10 + 2.10i)19-s + 3.78i·23-s + 1.00i·25-s + (−3.15 + 3.15i)29-s − 0.745i·31-s + (2.71 + 2.71i)35-s + (7.10 − 7.10i)37-s + 10.5·41-s + (2.04 + 2.04i)43-s − 10.9·47-s + ⋯
L(s)  = 1  + (−0.316 − 0.316i)5-s − 1.45·7-s + (0.0647 − 0.0647i)11-s + (−0.0493 − 0.0493i)13-s − 0.314i·17-s + (−0.483 + 0.483i)19-s + 0.789i·23-s + 0.200i·25-s + (−0.585 + 0.585i)29-s − 0.133i·31-s + (0.458 + 0.458i)35-s + (1.16 − 1.16i)37-s + 1.64·41-s + (0.311 + 0.311i)43-s − 1.59·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.991 - 0.127i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.991 - 0.127i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.112301973\)
\(L(\frac12)\) \(\approx\) \(1.112301973\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.83T + 7T^{2} \)
11 \( 1 + (-0.214 + 0.214i)T - 11iT^{2} \)
13 \( 1 + (0.177 + 0.177i)T + 13iT^{2} \)
17 \( 1 + 1.29iT - 17T^{2} \)
19 \( 1 + (2.10 - 2.10i)T - 19iT^{2} \)
23 \( 1 - 3.78iT - 23T^{2} \)
29 \( 1 + (3.15 - 3.15i)T - 29iT^{2} \)
31 \( 1 + 0.745iT - 31T^{2} \)
37 \( 1 + (-7.10 + 7.10i)T - 37iT^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + (-2.04 - 2.04i)T + 43iT^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + (-0.467 - 0.467i)T + 53iT^{2} \)
59 \( 1 + (-1.61 + 1.61i)T - 59iT^{2} \)
61 \( 1 + (-1.34 - 1.34i)T + 61iT^{2} \)
67 \( 1 + (-6.92 + 6.92i)T - 67iT^{2} \)
71 \( 1 - 8.46iT - 71T^{2} \)
73 \( 1 + 7.24iT - 73T^{2} \)
79 \( 1 + 12.5iT - 79T^{2} \)
83 \( 1 + (-9.42 - 9.42i)T + 83iT^{2} \)
89 \( 1 - 9.41T + 89T^{2} \)
97 \( 1 - 18.1T + 97T^{2} \)
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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

−9.037767123295645850517921874738, −7.903113255261182255875343197772, −7.37256372359985160692218180549, −6.39281928192790945401793878606, −5.89679285708731103234163448429, −4.89102235573335422900574095445, −3.86651245622117921218224204366, −3.28987167993740302386551193283, −2.18589876708340394374826537992, −0.68178482413872606623524211013, 0.57335304894822285056949103671, 2.28578348962525808763648646296, 3.09404359998019607205228444844, 3.92528020611817875641314325894, 4.73581933859671681626652294960, 6.00324477890798870556522234413, 6.41756334559148147248061653083, 7.14848907553725319185245722442, 7.988415355829113573016193321598, 8.800108098455200599796074874027

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