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Properties

Label 2-2880-60.59-c1-0-12
Degree $2$
Conductor $2880$
Sign $-0.479 - 0.877i$
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  + (−0.256 + 2.22i)5-s + 3.50·7-s + 1.92·11-s + 5.50i·13-s − 4.44·17-s + 7.00i·19-s − 1.10i·23-s + (−4.86 − 1.14i)25-s − 5.47i·29-s + 8.28i·31-s + (−0.900 + 7.78i)35-s + 0.778i·37-s + 2.44i·41-s − 9.55·43-s − 11.7i·47-s + ⋯
L(s)  = 1  + (−0.114 + 0.993i)5-s + 1.32·7-s + 0.581·11-s + 1.52i·13-s − 1.07·17-s + 1.60i·19-s − 0.229i·23-s + (−0.973 − 0.228i)25-s − 1.01i·29-s + 1.48i·31-s + (−0.152 + 1.31i)35-s + 0.127i·37-s + 0.381i·41-s − 1.45·43-s − 1.70i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.479 - 0.877i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2879, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.479 - 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.755007078\)
\(L(\frac12)\) \(\approx\) \(1.755007078\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.256 - 2.22i)T \)
good7 \( 1 - 3.50T + 7T^{2} \)
11 \( 1 - 1.92T + 11T^{2} \)
13 \( 1 - 5.50iT - 13T^{2} \)
17 \( 1 + 4.44T + 17T^{2} \)
19 \( 1 - 7.00iT - 19T^{2} \)
23 \( 1 + 1.10iT - 23T^{2} \)
29 \( 1 + 5.47iT - 29T^{2} \)
31 \( 1 - 8.28iT - 31T^{2} \)
37 \( 1 - 0.778iT - 37T^{2} \)
41 \( 1 - 2.44iT - 41T^{2} \)
43 \( 1 + 9.55T + 43T^{2} \)
47 \( 1 + 11.7iT - 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 + 9.78T + 59T^{2} \)
61 \( 1 - 3.45T + 61T^{2} \)
67 \( 1 - 5.45T + 67T^{2} \)
71 \( 1 + 4.25T + 71T^{2} \)
73 \( 1 + 7.27iT - 73T^{2} \)
79 \( 1 - 2.82iT - 79T^{2} \)
83 \( 1 - 4.25iT - 83T^{2} \)
89 \( 1 + 0.386iT - 89T^{2} \)
97 \( 1 + 9.29iT - 97T^{2} \)
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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

−8.816782606279706329220965978782, −8.343056533086717377870574244326, −7.44955950652169773573275203839, −6.73367247638813355701084455272, −6.19729630332156372534278358677, −5.04405640425003763506527093102, −4.23889573649674885173893538979, −3.58543866274235747217327715808, −2.17678543536411096798628560568, −1.62964820557094025374646080641, 0.55364345970415237778214291617, 1.57783681797168389372532059095, 2.67381126613234837659134202898, 3.94840261909059081266903774393, 4.76685275506592587241139012585, 5.19125293936912545012348468496, 6.11181211474102596830087133282, 7.21820322633005323662569070677, 7.85770309405100366142548843795, 8.614576912119548809380365644009

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