Properties

Label 2-2880-15.8-c1-0-45
Degree $2$
Conductor $2880$
Sign $-0.749 + 0.662i$
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  + (2.12 − 0.707i)5-s − 5.65i·11-s + (−3 − 3i)13-s − 4i·19-s + (−2.82 + 2.82i)23-s + (3.99 − 3i)25-s + 1.41·29-s − 8·31-s + (−7 + 7i)37-s − 1.41i·41-s + (−4 − 4i)43-s + (−2.82 − 2.82i)47-s + 7i·49-s + (−8.48 + 8.48i)53-s + (−4.00 − 12i)55-s + ⋯
L(s)  = 1  + (0.948 − 0.316i)5-s − 1.70i·11-s + (−0.832 − 0.832i)13-s − 0.917i·19-s + (−0.589 + 0.589i)23-s + (0.799 − 0.600i)25-s + 0.262·29-s − 1.43·31-s + (−1.15 + 1.15i)37-s − 0.220i·41-s + (−0.609 − 0.609i)43-s + (−0.412 − 0.412i)47-s + i·49-s + (−1.16 + 1.16i)53-s + (−0.539 − 1.61i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.749 + 0.662i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.297668609\)
\(L(\frac12)\) \(\approx\) \(1.297668609\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 + 0.707i)T \)
good7 \( 1 - 7iT^{2} \)
11 \( 1 + 5.65iT - 11T^{2} \)
13 \( 1 + (3 + 3i)T + 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (2.82 - 2.82i)T - 23iT^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (7 - 7i)T - 37iT^{2} \)
41 \( 1 + 1.41iT - 41T^{2} \)
43 \( 1 + (4 + 4i)T + 43iT^{2} \)
47 \( 1 + (2.82 + 2.82i)T + 47iT^{2} \)
53 \( 1 + (8.48 - 8.48i)T - 53iT^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 + (-8 + 8i)T - 67iT^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 + (3 + 3i)T + 73iT^{2} \)
79 \( 1 + 8iT - 79T^{2} \)
83 \( 1 + (11.3 - 11.3i)T - 83iT^{2} \)
89 \( 1 - 7.07T + 89T^{2} \)
97 \( 1 + (-5 + 5i)T - 97iT^{2} \)
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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

−8.579439092288309152090516584486, −7.83766549543443364303722333361, −6.88912266892462327946331583256, −6.04875020032366835358665743845, −5.44313364434606801545124964154, −4.82114643986369822028215117320, −3.46836789970035305268195261386, −2.79194143386596539272602371418, −1.62899641668849459845598898607, −0.37184448972186649717697009121, 1.88353560091855556645029996084, 2.06373058771165965089416450639, 3.45751092289754869993124149199, 4.50413201931727212126862812475, 5.15199038095200243812110491781, 6.06290941279953202443254619256, 6.93179376084952679061625924362, 7.28632171107719800268339154277, 8.342555142346870465979318118390, 9.283392584506873701152439271063

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