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Properties

Label 2-880-11.4-c1-0-1
Degree $2$
Conductor $880$
Sign $-0.999 + 0.00395i$
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  + (−0.655 + 2.01i)3-s + (0.809 − 0.587i)5-s + (−0.0946 − 0.291i)7-s + (−1.21 − 0.884i)9-s + (−2.72 + 1.89i)11-s + (−2.68 − 1.95i)13-s + (0.655 + 2.01i)15-s + (−4.58 + 3.33i)17-s + (−0.464 + 1.43i)19-s + 0.650·21-s + 0.343·23-s + (0.309 − 0.951i)25-s + (−2.56 + 1.86i)27-s + (2.15 + 6.64i)29-s + (−4.80 − 3.49i)31-s + ⋯
L(s)  = 1  + (−0.378 + 1.16i)3-s + (0.361 − 0.262i)5-s + (−0.0357 − 0.110i)7-s + (−0.405 − 0.294i)9-s + (−0.820 + 0.571i)11-s + (−0.745 − 0.541i)13-s + (0.169 + 0.521i)15-s + (−1.11 + 0.808i)17-s + (−0.106 + 0.328i)19-s + 0.141·21-s + 0.0716·23-s + (0.0618 − 0.190i)25-s + (−0.494 + 0.359i)27-s + (0.400 + 1.23i)29-s + (−0.863 − 0.627i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $-0.999 + 0.00395i$
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: \((2,\ 880,\ (\ :1/2),\ -0.999 + 0.00395i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00122742 - 0.619932i\)
\(L(\frac12)\) \(\approx\) \(0.00122742 - 0.619932i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.72 - 1.89i)T \)
good3 \( 1 + (0.655 - 2.01i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (0.0946 + 0.291i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.68 + 1.95i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.58 - 3.33i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.464 - 1.43i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 0.343T + 23T^{2} \)
29 \( 1 + (-2.15 - 6.64i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.80 + 3.49i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.63 - 5.04i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.25 - 6.94i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 4.16T + 43T^{2} \)
47 \( 1 + (-1.94 + 5.98i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (8.63 + 6.27i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.590 - 1.81i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (8.27 - 6.01i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 10.4T + 67T^{2} \)
71 \( 1 + (-9.03 + 6.56i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.792 + 2.43i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.95 - 1.42i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.66 + 2.66i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 2.46T + 89T^{2} \)
97 \( 1 + (11.1 + 8.06i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.41243538461290658481830767828, −9.936831157011614391650910052195, −9.093940753449519282460001403725, −8.153055751877517771621777281833, −7.14151938549471722883588905433, −6.01727575645440070828217315253, −5.02506210280135076867837148641, −4.59664344102306633169578775674, −3.38180649700657143863659282527, −2.00376000154456963888711008251, 0.28802360970746959906791391214, 1.95971175837046930014716861527, 2.80216734870516798793046712332, 4.45892606660809449956476477310, 5.53511989464766170654969055578, 6.37566230455774823185720357889, 7.11909870531731466384954114392, 7.74856209544487804931098401105, 8.887697218946953342946203976823, 9.647488936837829380543902266728

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