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Properties

Label 2-880-11.3-c1-0-12
Degree $2$
Conductor $880$
Sign $0.602 - 0.798i$
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  + (0.346 + 1.06i)3-s + (0.809 + 0.587i)5-s + (−0.714 + 2.19i)7-s + (1.40 − 1.02i)9-s + (3.14 − 1.04i)11-s + (4.18 − 3.04i)13-s + (−0.346 + 1.06i)15-s + (−0.339 − 0.246i)17-s + (0.537 + 1.65i)19-s − 2.59·21-s − 4.90·23-s + (0.309 + 0.951i)25-s + (4.30 + 3.12i)27-s + (−0.849 + 2.61i)29-s + (5.30 − 3.85i)31-s + ⋯
L(s)  = 1  + (0.200 + 0.616i)3-s + (0.361 + 0.262i)5-s + (−0.269 + 0.830i)7-s + (0.469 − 0.340i)9-s + (0.949 − 0.314i)11-s + (1.16 − 0.843i)13-s + (−0.0895 + 0.275i)15-s + (−0.0824 − 0.0598i)17-s + (0.123 + 0.379i)19-s − 0.566·21-s − 1.02·23-s + (0.0618 + 0.190i)25-s + (0.828 + 0.601i)27-s + (−0.157 + 0.485i)29-s + (0.953 − 0.692i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 - 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.602 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.602 - 0.798i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ 0.602 - 0.798i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76796 + 0.880842i\)
\(L(\frac12)\) \(\approx\) \(1.76796 + 0.880842i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.14 + 1.04i)T \)
good3 \( 1 + (-0.346 - 1.06i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (0.714 - 2.19i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-4.18 + 3.04i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.339 + 0.246i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.537 - 1.65i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.90T + 23T^{2} \)
29 \( 1 + (0.849 - 2.61i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.30 + 3.85i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.22 - 9.93i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.61 - 8.03i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.16T + 43T^{2} \)
47 \( 1 + (-0.320 - 0.987i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.38 - 3.18i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-4.45 + 13.7i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (10.2 + 7.47i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 0.559T + 67T^{2} \)
71 \( 1 + (-4.17 - 3.03i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.83 + 11.8i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (3.91 - 2.84i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.83 + 4.23i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 8.95T + 89T^{2} \)
97 \( 1 + (-0.633 + 0.460i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.04721621238622603671022396150, −9.547593543190369001961479247826, −8.698386223640011632346662211423, −7.964991312296211708599258625235, −6.40739153528272700999527656016, −6.18409752062774278798104618621, −4.92182475750407829967861450176, −3.74637714445379507322026566931, −3.04512610580591856301749002168, −1.44500750185388446154747188732, 1.14727054069634204714952138795, 2.09752194086402870848673430361, 3.79705852286136222272326997050, 4.41958060251583771673051182104, 5.86384050867118117760665871951, 6.75791840919519224498216430070, 7.25848304551099114427431676156, 8.381346249042781528370436697183, 9.107944034420064913034598772918, 10.02955051871186628278887123002

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