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Properties

Label 2-2880-5.4-c1-0-54
Degree $2$
Conductor $2880$
Sign $-0.894 + 0.447i$
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  + (2 − i)5-s − 2i·7-s − 2·11-s − 2i·13-s + 6i·17-s − 8·19-s − 4i·23-s + (3 − 4i)25-s − 8·29-s + (−2 − 4i)35-s − 10i·37-s − 2·41-s − 12i·43-s + 3·49-s + 10i·53-s + ⋯
L(s)  = 1  + (0.894 − 0.447i)5-s − 0.755i·7-s − 0.603·11-s − 0.554i·13-s + 1.45i·17-s − 1.83·19-s − 0.834i·23-s + (0.600 − 0.800i)25-s − 1.48·29-s + (−0.338 − 0.676i)35-s − 1.64i·37-s − 0.312·41-s − 1.82i·43-s + 0.428·49-s + 1.37i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9513136281\)
\(L(\frac12)\) \(\approx\) \(0.9513136281\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + i)T \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 12iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−8.587390452096719113575489504736, −7.74239352074784026533106070803, −6.91979212716922612287566611429, −5.98311954980599373297149254192, −5.57009367277100944341182535334, −4.41696931555720389855660790343, −3.84826378780299570183105465452, −2.46382969019994924043429099589, −1.69861728057308960598431550052, −0.26910078834385321988759637790, 1.71037523096834087919420244179, 2.46447590113798284131579703632, 3.27041264652413628877795841759, 4.59781379018489184756234385581, 5.28216964821954324955788193451, 6.09704490451645297475332832255, 6.69090216278290355321555768848, 7.54580330778737615263910704726, 8.423832746211050667883485286644, 9.268177394708323029039873049736

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