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Properties

Label 2-880-11.3-c1-0-8
Degree $2$
Conductor $880$
Sign $0.999 + 0.0206i$
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  + (−0.529 − 1.62i)3-s + (0.809 + 0.587i)5-s + (−1.14 + 3.52i)7-s + (0.0536 − 0.0389i)9-s + (−2.68 − 1.94i)11-s + (0.952 − 0.692i)13-s + (0.529 − 1.62i)15-s + (4.36 + 3.17i)17-s + (1.18 + 3.65i)19-s + 6.34·21-s + 8.68·23-s + (0.309 + 0.951i)25-s + (−4.24 − 3.08i)27-s + (−2.12 + 6.53i)29-s + (7.08 − 5.14i)31-s + ⋯
L(s)  = 1  + (−0.305 − 0.940i)3-s + (0.361 + 0.262i)5-s + (−0.432 + 1.33i)7-s + (0.0178 − 0.0129i)9-s + (−0.810 − 0.585i)11-s + (0.264 − 0.191i)13-s + (0.136 − 0.420i)15-s + (1.05 + 0.769i)17-s + (0.272 + 0.839i)19-s + 1.38·21-s + 1.81·23-s + (0.0618 + 0.190i)25-s + (−0.817 − 0.594i)27-s + (−0.394 + 1.21i)29-s + (1.27 − 0.924i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0206i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.999 + 0.0206i$
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: \((2,\ 880,\ (\ :1/2),\ 0.999 + 0.0206i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46702 - 0.0151531i\)
\(L(\frac12)\) \(\approx\) \(1.46702 - 0.0151531i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.68 + 1.94i)T \)
good3 \( 1 + (0.529 + 1.62i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (1.14 - 3.52i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.952 + 0.692i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.36 - 3.17i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.18 - 3.65i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 8.68T + 23T^{2} \)
29 \( 1 + (2.12 - 6.53i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-7.08 + 5.14i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.696 + 2.14i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.493 - 1.51i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 4.11T + 43T^{2} \)
47 \( 1 + (-3.91 - 12.0i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-10.0 + 7.27i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.121 + 0.374i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.45 + 1.05i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 14.3T + 67T^{2} \)
71 \( 1 + (5.54 + 4.02i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.10 + 9.56i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.901 - 0.654i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.0140 - 0.0101i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 8.49T + 89T^{2} \)
97 \( 1 + (8.50 - 6.18i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.12928589997160058287104293080, −9.279739187375656034973836881718, −8.347298224645001258267515195103, −7.59424945243083614328021437332, −6.53321568154535794901707201465, −5.86282266250004501275117678070, −5.27873287334432058356071670836, −3.43291701645815627572933881042, −2.52854666005754416282488818944, −1.19565574960920464737606746182, 0.911416885261561487896070151605, 2.81207590913558847250912048115, 3.96195757863562109447506473000, 4.83868485731211605093477501688, 5.42851477283673468607801903771, 6.92560663873292625204511944421, 7.35753528003260948837647022914, 8.613663049774648429577697359287, 9.771342812357913233112602303310, 9.977594936828869547794769026406

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