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Properties

Label 2-880-11.3-c1-0-17
Degree $2$
Conductor $880$
Sign $0.440 + 0.897i$
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  + (−0.177 − 0.547i)3-s + (0.809 + 0.587i)5-s + (1.12 − 3.47i)7-s + (2.15 − 1.56i)9-s + (−0.490 + 3.28i)11-s + (2.29 − 1.66i)13-s + (0.177 − 0.547i)15-s + (−2.98 − 2.17i)17-s + (0.0293 + 0.0904i)19-s − 2.10·21-s − 1.16·23-s + (0.309 + 0.951i)25-s + (−2.63 − 1.91i)27-s + (−2.08 + 6.42i)29-s + (5.48 − 3.98i)31-s + ⋯
L(s)  = 1  + (−0.102 − 0.315i)3-s + (0.361 + 0.262i)5-s + (0.426 − 1.31i)7-s + (0.719 − 0.522i)9-s + (−0.147 + 0.989i)11-s + (0.635 − 0.461i)13-s + (0.0459 − 0.141i)15-s + (−0.724 − 0.526i)17-s + (0.00674 + 0.0207i)19-s − 0.458·21-s − 0.242·23-s + (0.0618 + 0.190i)25-s + (−0.507 − 0.369i)27-s + (−0.387 + 1.19i)29-s + (0.984 − 0.715i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.440 + 0.897i$
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: \((2,\ 880,\ (\ :1/2),\ 0.440 + 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46676 - 0.914109i\)
\(L(\frac12)\) \(\approx\) \(1.46676 - 0.914109i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.490 - 3.28i)T \)
good3 \( 1 + (0.177 + 0.547i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-1.12 + 3.47i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.29 + 1.66i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.98 + 2.17i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.0293 - 0.0904i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.16T + 23T^{2} \)
29 \( 1 + (2.08 - 6.42i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.48 + 3.98i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.04 + 9.35i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.57 + 7.91i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.96T + 43T^{2} \)
47 \( 1 + (-0.687 - 2.11i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.42 - 1.75i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.62 + 8.09i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.86 - 4.98i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 + (-6.71 - 4.88i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.407 - 1.25i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (11.2 - 8.15i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (8.61 + 6.25i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 + (-3.50 + 2.54i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.05199773497268518947514493141, −9.338167459705094174494984031632, −8.137513112604179252660591332017, −7.13316100776908285648171078709, −6.92960765574655250759325954404, −5.66875872823725525761419946800, −4.47782914483702712848969602528, −3.76598887600749613638144355664, −2.17257794492365013591415070593, −0.930607650645638655614440456585, 1.57615467347272262414299914567, 2.68833115532689616752372706878, 4.12604934046040278654595024825, 5.02915153711191487162313797489, 5.89255702661979190396490562115, 6.61297193975791982817119398896, 8.184762636900511430436484363782, 8.463988077818216626034623566206, 9.460514924610377323021031378390, 10.21385246144377640521972443392

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