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Properties

Label 2-1120-56.27-c1-0-6
Degree $2$
Conductor $1120$
Sign $0.594 - 0.804i$
Analytic cond. $8.94324$
Root an. cond. $2.99052$
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  − 0.586i·3-s − 5-s + (−2.52 + 0.792i)7-s + 2.65·9-s + 0.580·11-s − 1.14·13-s + 0.586i·15-s + 1.82i·17-s + 4.72i·19-s + (0.464 + 1.48i)21-s − 2.79i·23-s + 25-s − 3.31i·27-s + 7.40i·29-s + 4.73·31-s + ⋯
L(s)  = 1  − 0.338i·3-s − 0.447·5-s + (−0.954 + 0.299i)7-s + 0.885·9-s + 0.174·11-s − 0.318·13-s + 0.151i·15-s + 0.443i·17-s + 1.08i·19-s + (0.101 + 0.323i)21-s − 0.583i·23-s + 0.200·25-s − 0.638i·27-s + 1.37i·29-s + 0.849·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.594 - 0.804i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ 0.594 - 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.204839234\)
\(L(\frac12)\) \(\approx\) \(1.204839234\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + (2.52 - 0.792i)T \)
good3 \( 1 + 0.586iT - 3T^{2} \)
11 \( 1 - 0.580T + 11T^{2} \)
13 \( 1 + 1.14T + 13T^{2} \)
17 \( 1 - 1.82iT - 17T^{2} \)
19 \( 1 - 4.72iT - 19T^{2} \)
23 \( 1 + 2.79iT - 23T^{2} \)
29 \( 1 - 7.40iT - 29T^{2} \)
31 \( 1 - 4.73T + 31T^{2} \)
37 \( 1 - 4.35iT - 37T^{2} \)
41 \( 1 - 8.46iT - 41T^{2} \)
43 \( 1 - 4.32T + 43T^{2} \)
47 \( 1 - 3.56T + 47T^{2} \)
53 \( 1 - 5.48iT - 53T^{2} \)
59 \( 1 - 13.2iT - 59T^{2} \)
61 \( 1 + 0.275T + 61T^{2} \)
67 \( 1 - 7.71T + 67T^{2} \)
71 \( 1 - 1.13iT - 71T^{2} \)
73 \( 1 + 1.78iT - 73T^{2} \)
79 \( 1 - 10.0iT - 79T^{2} \)
83 \( 1 + 15.6iT - 83T^{2} \)
89 \( 1 + 15.3iT - 89T^{2} \)
97 \( 1 - 14.9iT - 97T^{2} \)
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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

−10.05067687575390380789845181291, −9.138696895314633270666126701129, −8.266879416213247893615624842669, −7.40686648342370804956517948936, −6.64725943061961650478432898893, −5.92843820217229179411811794012, −4.65125656014105742389619183149, −3.76086227499239869710616335553, −2.70573200928007615546360972712, −1.26076143360306152201734627811, 0.59669716054952721708160606038, 2.43881208586371189270119880874, 3.63249578522031647035515901860, 4.31957762797833944424699466743, 5.32889302212455666283713164741, 6.56128326878326613928239939872, 7.14096753958302269813624966410, 7.955614815659542559332583949438, 9.204401318725577013536648637934, 9.613582943652091564318845149675

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