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Properties

Label 2-2880-48.35-c1-0-30
Degree $2$
Conductor $2880$
Sign $-0.855 + 0.517i$
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 + 0.527·7-s + (−1.36 + 1.36i)11-s + (−2.79 − 2.79i)13-s − 6.68i·17-s + (−3.79 + 3.79i)19-s + 5.00i·23-s + 1.00i·25-s + (2.94 − 2.94i)29-s − 6.21i·31-s + (0.372 + 0.372i)35-s + (−6.42 + 6.42i)37-s + 2.16·41-s + (−4.71 − 4.71i)43-s − 11.3·47-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s + 0.199·7-s + (−0.412 + 0.412i)11-s + (−0.774 − 0.774i)13-s − 1.62i·17-s + (−0.870 + 0.870i)19-s + 1.04i·23-s + 0.200i·25-s + (0.546 − 0.546i)29-s − 1.11i·31-s + (0.0630 + 0.0630i)35-s + (−1.05 + 1.05i)37-s + 0.337·41-s + (−0.719 − 0.719i)43-s − 1.65·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.855 + 0.517i$
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: \((2,\ 2880,\ (\ :1/2),\ -0.855 + 0.517i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3977790912\)
\(L(\frac12)\) \(\approx\) \(0.3977790912\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.527T + 7T^{2} \)
11 \( 1 + (1.36 - 1.36i)T - 11iT^{2} \)
13 \( 1 + (2.79 + 2.79i)T + 13iT^{2} \)
17 \( 1 + 6.68iT - 17T^{2} \)
19 \( 1 + (3.79 - 3.79i)T - 19iT^{2} \)
23 \( 1 - 5.00iT - 23T^{2} \)
29 \( 1 + (-2.94 + 2.94i)T - 29iT^{2} \)
31 \( 1 + 6.21iT - 31T^{2} \)
37 \( 1 + (6.42 - 6.42i)T - 37iT^{2} \)
41 \( 1 - 2.16T + 41T^{2} \)
43 \( 1 + (4.71 + 4.71i)T + 43iT^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 + (4.23 + 4.23i)T + 53iT^{2} \)
59 \( 1 + (-0.322 + 0.322i)T - 59iT^{2} \)
61 \( 1 + (0.887 + 0.887i)T + 61iT^{2} \)
67 \( 1 + (1.24 - 1.24i)T - 67iT^{2} \)
71 \( 1 + 14.3iT - 71T^{2} \)
73 \( 1 - 7.97iT - 73T^{2} \)
79 \( 1 - 5.37iT - 79T^{2} \)
83 \( 1 + (-2.43 - 2.43i)T + 83iT^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 - 2.81T + 97T^{2} \)
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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

−8.256066334419559425126619258949, −7.76661159401661869476014232362, −6.99206938917833895708490201117, −6.20625448542814387213678010189, −5.21887632475781699609873832241, −4.78487663648520425143895519251, −3.51580165152463577203367227437, −2.67279387660970588514666945582, −1.75912132072982252482495759337, −0.11620262459572784414897250002, 1.52074717474840191731197936943, 2.39906090688664179453453722932, 3.49863626421226203835995536671, 4.61412845139101730308577500643, 5.01412383579038311104814721415, 6.22204467578488570912251407622, 6.62677078604149579977723115490, 7.64277707451948126934434559726, 8.610425642945300039288901486863, 8.769633111649201247901517454964

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