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Properties

Label 2-1120-35.24-c0-0-3
Degree $2$
Conductor $1120$
Sign $-0.134 + 0.990i$
Analytic cond. $0.558952$
Root an. cond. $0.747631$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 − 1.67i)3-s + (−0.866 + 0.5i)5-s + (0.965 − 0.258i)7-s + (−1.36 − 2.36i)9-s + 1.93i·15-s + (0.500 − 1.86i)21-s + (0.448 − 0.258i)23-s + (0.499 − 0.866i)25-s − 3.34·27-s − 29-s + (−0.707 + 0.707i)35-s + i·41-s + 0.517i·43-s + (2.36 + 1.36i)45-s + (0.707 + 1.22i)47-s + ⋯
L(s)  = 1  + (0.965 − 1.67i)3-s + (−0.866 + 0.5i)5-s + (0.965 − 0.258i)7-s + (−1.36 − 2.36i)9-s + 1.93i·15-s + (0.500 − 1.86i)21-s + (0.448 − 0.258i)23-s + (0.499 − 0.866i)25-s − 3.34·27-s − 29-s + (−0.707 + 0.707i)35-s + i·41-s + 0.517i·43-s + (2.36 + 1.36i)45-s + (0.707 + 1.22i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $-0.134 + 0.990i$
Analytic conductor: \(0.558952\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :0),\ -0.134 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.292784751\)
\(L(\frac12)\) \(\approx\) \(1.292784751\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-0.965 + 0.258i)T \)
good3 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.448 + 0.258i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - iT - T^{2} \)
43 \( 1 - 0.517iT - T^{2} \)
47 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - 0.517T + T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.554612858475464407520555476553, −8.554758898569440684140231908438, −8.068518890450694213545149675236, −7.39006648431917864079533645631, −6.88726062395558982027481415756, −5.85334316921964232122020023108, −4.36583385957494291518149300929, −3.29338415571379948530706807376, −2.37202897412894932319459079457, −1.18435111923841748470881495417, 2.13691421231013788083692136969, 3.42676679760767010258606071557, 4.07706761165233261892292319570, 4.93397630462943580967068106471, 5.47360517434381524862644284497, 7.38202914513482818187544225419, 8.059722833839157799642822654791, 8.785563386057990227058256233976, 9.193641455938302466386257994958, 10.20286602380315597405945158351

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