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Properties

Label 2-2880-5.3-c0-0-1
Degree $2$
Conductor $2880$
Sign $0.525 - 0.850i$
Analytic cond. $1.43730$
Root an. cond. $1.19887$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·5-s + (1 + i)13-s + (1 − i)17-s − 25-s + (−1 + i)37-s + i·49-s + (1 + i)53-s + (−1 + i)65-s + (1 + i)73-s + (1 + i)85-s + (1 − i)97-s − 2·101-s − 2i·109-s + (1 + i)113-s + ⋯
L(s)  = 1  + i·5-s + (1 + i)13-s + (1 − i)17-s − 25-s + (−1 + i)37-s + i·49-s + (1 + i)53-s + (−1 + i)65-s + (1 + i)73-s + (1 + i)85-s + (1 − i)97-s − 2·101-s − 2i·109-s + (1 + i)113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.283631213\)
\(L(\frac12)\) \(\approx\) \(1.283631213\)
\(L(1)\) 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 \( 1 - iT \)
good7 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-1 - i)T + iT^{2} \)
17 \( 1 + (-1 + i)T - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (1 - i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-1 - i)T + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1 + i)T - iT^{2} \)
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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

−9.128242467671934271318311889394, −8.298600586249430583897257021157, −7.44195315300151544239062025821, −6.84099824503095548871437849223, −6.14375428429143603841538371904, −5.32354147009480676984849420385, −4.23837046942008213553214128633, −3.42414984652996659659461649182, −2.62481241911682814146097580694, −1.41376208015729608012874519350, 0.917948674155859498351291349617, 1.96688198360800939298263431269, 3.42529397274657834581252151177, 3.95216783203798512716692098249, 5.16568764777062778996810697081, 5.61163699486104695045117264255, 6.41902427963126725495844519235, 7.53301541283942546343126999309, 8.236205154807722927687270675691, 8.652418937786704961688267689441

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