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Properties

Label 2-880-11.4-c1-0-13
Degree $2$
Conductor $880$
Sign $0.995 - 0.0913i$
Analytic cond. $7.02683$
Root an. cond. $2.65081$
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  + (−1 + 3.07i)3-s + (0.809 − 0.587i)5-s + (−0.809 − 2.48i)7-s + (−6.04 − 4.39i)9-s + (2.54 − 2.12i)11-s + (1.30 + 0.951i)13-s + (1 + 3.07i)15-s + (4.23 − 3.07i)17-s + (1.26 − 3.88i)19-s + 8.47·21-s − 0.145·23-s + (0.309 − 0.951i)25-s + (11.7 − 8.50i)27-s + (−0.381 − 1.17i)29-s + (−5.85 − 4.25i)31-s + ⋯
L(s)  = 1  + (−0.577 + 1.77i)3-s + (0.361 − 0.262i)5-s + (−0.305 − 0.941i)7-s + (−2.01 − 1.46i)9-s + (0.767 − 0.641i)11-s + (0.363 + 0.263i)13-s + (0.258 + 0.794i)15-s + (1.02 − 0.746i)17-s + (0.289 − 0.892i)19-s + 1.84·21-s − 0.0304·23-s + (0.0618 − 0.190i)25-s + (2.25 − 1.63i)27-s + (−0.0709 − 0.218i)29-s + (−1.05 − 0.763i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.995 - 0.0913i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ 0.995 - 0.0913i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24439 + 0.0569826i\)
\(L(\frac12)\) \(\approx\) \(1.24439 + 0.0569826i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-2.54 + 2.12i)T \)
good3 \( 1 + (1 - 3.07i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (0.809 + 2.48i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.30 - 0.951i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.23 + 3.07i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.26 + 3.88i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 0.145T + 23T^{2} \)
29 \( 1 + (0.381 + 1.17i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.85 + 4.25i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.263 + 0.812i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.572 - 1.76i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 9.23T + 43T^{2} \)
47 \( 1 + (3.5 - 10.7i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.736 + 0.534i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.736 + 2.26i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (7.23 - 5.25i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 0.763T + 67T^{2} \)
71 \( 1 + (-10.7 + 7.77i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.527 + 1.62i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.23 + 0.898i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-12.7 + 9.23i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 + (-9.70 - 7.05i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.09921917412389695948015045490, −9.353490689155568269925164976311, −9.046215560547240836116013176611, −7.62852140784244388913173795429, −6.38402968599065687130744263661, −5.65694633471211819081122103975, −4.71616291776267492174288850347, −3.92269577987806202708512371254, −3.13838019675347849776470100700, −0.72729118320162905364509830014, 1.33337557006603995058735957164, 2.16799708877162877619198917754, 3.45877867266554865598593383211, 5.39713861190806417640492861228, 5.88874120244476650114791589831, 6.64381204638934028287994536325, 7.42881956736820521785702832750, 8.258623918654078598976694571515, 9.132610803103843890836462062047, 10.23090858598266857128729597208

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