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Properties

Label 2-2880-48.35-c1-0-27
Degree $2$
Conductor $2880$
Sign $0.999 + 0.0438i$
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 + 3.46·7-s + (2.99 − 2.99i)11-s + (4.77 + 4.77i)13-s − 2.39i·17-s + (3.53 − 3.53i)19-s − 0.278i·23-s + 1.00i·25-s + (3.01 − 3.01i)29-s − 0.996i·31-s + (2.45 + 2.45i)35-s + (−7.22 + 7.22i)37-s − 2.39·41-s + (−6.32 − 6.32i)43-s + 4.37·47-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s + 1.31·7-s + (0.903 − 0.903i)11-s + (1.32 + 1.32i)13-s − 0.579i·17-s + (0.809 − 0.809i)19-s − 0.0581i·23-s + 0.200i·25-s + (0.559 − 0.559i)29-s − 0.178i·31-s + (0.414 + 0.414i)35-s + (−1.18 + 1.18i)37-s − 0.374·41-s + (−0.965 − 0.965i)43-s + 0.638·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.999 + 0.0438i$
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: \((2,\ 2880,\ (\ :1/2),\ 0.999 + 0.0438i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.723950110\)
\(L(\frac12)\) \(\approx\) \(2.723950110\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.46T + 7T^{2} \)
11 \( 1 + (-2.99 + 2.99i)T - 11iT^{2} \)
13 \( 1 + (-4.77 - 4.77i)T + 13iT^{2} \)
17 \( 1 + 2.39iT - 17T^{2} \)
19 \( 1 + (-3.53 + 3.53i)T - 19iT^{2} \)
23 \( 1 + 0.278iT - 23T^{2} \)
29 \( 1 + (-3.01 + 3.01i)T - 29iT^{2} \)
31 \( 1 + 0.996iT - 31T^{2} \)
37 \( 1 + (7.22 - 7.22i)T - 37iT^{2} \)
41 \( 1 + 2.39T + 41T^{2} \)
43 \( 1 + (6.32 + 6.32i)T + 43iT^{2} \)
47 \( 1 - 4.37T + 47T^{2} \)
53 \( 1 + (7.49 + 7.49i)T + 53iT^{2} \)
59 \( 1 + (-7.26 + 7.26i)T - 59iT^{2} \)
61 \( 1 + (0.978 + 0.978i)T + 61iT^{2} \)
67 \( 1 + (4.87 - 4.87i)T - 67iT^{2} \)
71 \( 1 - 13.1iT - 71T^{2} \)
73 \( 1 - 14.2iT - 73T^{2} \)
79 \( 1 + 2.98iT - 79T^{2} \)
83 \( 1 + (11.2 + 11.2i)T + 83iT^{2} \)
89 \( 1 + 5.54T + 89T^{2} \)
97 \( 1 - 13.0T + 97T^{2} \)
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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

−8.602465178817668613985528897014, −8.324977631460802456040253625076, −7.03337629676307607663769156741, −6.64175030399597745968874995881, −5.68883930767657267341978345074, −4.88746431909957963215102565492, −4.03199556087466592411652611889, −3.16006515984653145433908203103, −1.88448239442704855986501601605, −1.10340076920937700865457123329, 1.26207311497121534351141314938, 1.70301458161691013531830754632, 3.22010740281579174829970784390, 4.07532574826271189231019528550, 4.96346802587207090266165771535, 5.63407836187355078046119970325, 6.39712287856345702111691880489, 7.45039868880853602187753718233, 8.078304174164177268110821922758, 8.686131005675622810409138750859

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