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Properties

Label 2-880-11.4-c1-0-14
Degree $2$
Conductor $880$
Sign $0.859 + 0.511i$
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  + (0.118 − 0.363i)3-s + (0.809 − 0.587i)5-s + (2.30 + 1.67i)9-s + (3.30 − 0.224i)11-s + (−5.23 − 3.80i)13-s + (−0.118 − 0.363i)15-s + (4.11 − 2.99i)17-s + (−1.19 + 3.66i)19-s + 6·23-s + (0.309 − 0.951i)25-s + (1.80 − 1.31i)27-s + (1.47 + 4.53i)29-s + (−3 − 2.17i)31-s + (0.309 − 1.22i)33-s + (−0.618 − 1.90i)37-s + ⋯
L(s)  = 1  + (0.0681 − 0.209i)3-s + (0.361 − 0.262i)5-s + (0.769 + 0.559i)9-s + (0.997 − 0.0676i)11-s + (−1.45 − 1.05i)13-s + (−0.0304 − 0.0937i)15-s + (0.998 − 0.725i)17-s + (−0.273 + 0.840i)19-s + 1.25·23-s + (0.0618 − 0.190i)25-s + (0.348 − 0.252i)27-s + (0.273 + 0.841i)29-s + (−0.538 − 0.391i)31-s + (0.0537 − 0.213i)33-s + (−0.101 − 0.312i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.859 + 0.511i$
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: \((2,\ 880,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79026 - 0.492387i\)
\(L(\frac12)\) \(\approx\) \(1.79026 - 0.492387i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.30 + 0.224i)T \)
good3 \( 1 + (-0.118 + 0.363i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (5.23 + 3.80i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.11 + 2.99i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.19 - 3.66i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + (-1.47 - 4.53i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3 + 2.17i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.618 + 1.90i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.5 - 4.61i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.85T + 43T^{2} \)
47 \( 1 + (-3.47 + 10.6i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.85 - 2.07i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.57 + 10.9i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.23 + 4.53i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 0.618T + 67T^{2} \)
71 \( 1 + (3.85 - 2.80i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.354 - 1.08i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.85 - 4.25i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (10.0 - 7.27i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 4.09T + 89T^{2} \)
97 \( 1 + (-6.16 - 4.47i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.985830273784157752158304073538, −9.394769241688187190083785931867, −8.324610021581143441510007155544, −7.43049768342632571979543970457, −6.84647289865728509663038842366, −5.49603533428830808868138786612, −4.92949968654195410395236448137, −3.63936259560345730915493385973, −2.39462182435982885377145105230, −1.08979403215113393851802691039, 1.35712205292447136471387394509, 2.70086950652882444142206156718, 3.98033003017654160826917477038, 4.71899654062276273011647475013, 5.95972398377433460588965353262, 6.93974912229450638271661907502, 7.34448742798484318129203209587, 8.869798628568234930835120767308, 9.381217945090375824689547705858, 10.06481512918152400173677216852

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