Properties

Label 2-2880-3.2-c2-0-47
Degree $2$
Conductor $2880$
Sign $-0.816 + 0.577i$
Analytic cond. $78.4743$
Root an. cond. $8.85857$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.23i·5-s − 5.48·7-s + 9.17i·11-s − 11.4·13-s + 16.9i·17-s + 26.9·19-s − 4.93i·23-s − 5.00·25-s + 20.5i·29-s − 20.9·31-s + 12.2i·35-s + 62.4·37-s − 40.9i·41-s + 1.02·43-s − 86.2i·47-s + ⋯
L(s)  = 1  − 0.447i·5-s − 0.783·7-s + 0.833i·11-s − 0.883·13-s + 0.998i·17-s + 1.41·19-s − 0.214i·23-s − 0.200·25-s + 0.707i·29-s − 0.676·31-s + 0.350i·35-s + 1.68·37-s − 0.998i·41-s + 0.0238·43-s − 1.83i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.816 + 0.577i$
Analytic conductor: \(78.4743\)
Root analytic conductor: \(8.85857\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1),\ -0.816 + 0.577i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4014850310\)
\(L(\frac12)\) \(\approx\) \(0.4014850310\)
\(L(2)\) 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 \( 1 + 2.23iT \)
good7 \( 1 + 5.48T + 49T^{2} \)
11 \( 1 - 9.17iT - 121T^{2} \)
13 \( 1 + 11.4T + 169T^{2} \)
17 \( 1 - 16.9iT - 289T^{2} \)
19 \( 1 - 26.9T + 361T^{2} \)
23 \( 1 + 4.93iT - 529T^{2} \)
29 \( 1 - 20.5iT - 841T^{2} \)
31 \( 1 + 20.9T + 961T^{2} \)
37 \( 1 - 62.4T + 1.36e3T^{2} \)
41 \( 1 + 40.9iT - 1.68e3T^{2} \)
43 \( 1 - 1.02T + 1.84e3T^{2} \)
47 \( 1 + 86.2iT - 2.20e3T^{2} \)
53 \( 1 - 96.0iT - 2.80e3T^{2} \)
59 \( 1 + 112. iT - 3.48e3T^{2} \)
61 \( 1 - 66.9T + 3.72e3T^{2} \)
67 \( 1 + 76T + 4.48e3T^{2} \)
71 \( 1 - 24.0iT - 5.04e3T^{2} \)
73 \( 1 + 18.9T + 5.32e3T^{2} \)
79 \( 1 + 106.T + 6.24e3T^{2} \)
83 \( 1 - 45.1iT - 6.88e3T^{2} \)
89 \( 1 + 115. iT - 7.92e3T^{2} \)
97 \( 1 + 87.0T + 9.40e3T^{2} \)
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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

−8.303959121798696387140423210330, −7.41520660317352841848694473374, −6.94732013537013690587582198102, −5.91314475412389150952921207072, −5.22786533927295692713409910925, −4.36160006421020878300760201664, −3.49575094875659462259462389557, −2.50397431913309075524282700028, −1.43123494579099588369557609827, −0.10156633301517625104581771976, 1.05051438311073317238814767513, 2.67862726336953698913511559911, 3.03783036431720831154404217285, 4.10377800838640465868986811766, 5.14125440920176012543193124167, 5.88397787306025640520337871543, 6.63387482867447840343869769282, 7.45227310828273580907988945858, 7.928362133050656001372736104237, 9.130263627346863343207916798823

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