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Properties

Label 2-2880-120.53-c1-0-47
Degree $2$
Conductor $2880$
Sign $-0.884 - 0.465i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.884 - 0.465i$
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: \((2,\ 2880,\ (\ :1/2),\ -0.884 - 0.465i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1489745934\)
\(L(\frac12)\) \(\approx\) \(0.1489745934\)
\(L(\frac{3}{2})\) 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

−8.621755667667818827809951648741, −7.35237523721695156976888201955, −7.08079501034542135765680981886, −5.99816963413140433737907171462, −5.07785952613106288412149449370, −4.35268338061888549956920646722, −3.88613134505969466064773941297, −2.20611087276317100029949060142, −1.52049118314154666225023605864, −0.04280567649166713338200761427, 2.02544721765024919866632819350, 2.37230429342180420815151817372, 3.79284579737281476289658251132, 4.32285071386698714719066899293, 5.52071842983198916472727766238, 6.33690834423923318207379301250, 6.70541670993988938876199501606, 7.83584051187059095843669082168, 8.355417005106853904884331680230, 9.163816895786845973644711346004

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