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Properties

Label 2-1120-56.3-c1-0-12
Degree $2$
Conductor $1120$
Sign $0.638 + 0.769i$
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  + (−2.26 − 1.30i)3-s + (0.5 + 0.866i)5-s + (1.72 − 2.00i)7-s + (1.92 + 3.33i)9-s + (−0.530 + 0.919i)11-s − 0.831·13-s − 2.61i·15-s + (4.14 + 2.39i)17-s + (2.03 − 1.17i)19-s + (−6.53 + 2.28i)21-s + (1.32 − 0.763i)23-s + (−0.499 + 0.866i)25-s − 2.23i·27-s + 3.07i·29-s + (−4.78 + 8.28i)31-s + ⋯
L(s)  = 1  + (−1.30 − 0.755i)3-s + (0.223 + 0.387i)5-s + (0.652 − 0.757i)7-s + (0.642 + 1.11i)9-s + (−0.160 + 0.277i)11-s − 0.230·13-s − 0.675i·15-s + (1.00 + 0.580i)17-s + (0.467 − 0.269i)19-s + (−1.42 + 0.498i)21-s + (0.275 − 0.159i)23-s + (−0.0999 + 0.173i)25-s − 0.429i·27-s + 0.571i·29-s + (−0.859 + 1.48i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.638 + 0.769i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ 0.638 + 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.131818536\)
\(L(\frac12)\) \(\approx\) \(1.131818536\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-1.72 + 2.00i)T \)
good3 \( 1 + (2.26 + 1.30i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (0.530 - 0.919i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.831T + 13T^{2} \)
17 \( 1 + (-4.14 - 2.39i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.03 + 1.17i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.32 + 0.763i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.07iT - 29T^{2} \)
31 \( 1 + (4.78 - 8.28i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-10.0 + 5.81i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.76iT - 41T^{2} \)
43 \( 1 - 4.99T + 43T^{2} \)
47 \( 1 + (1.75 + 3.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.61 - 3.81i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.31 - 1.91i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.51 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.36 + 12.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.99iT - 71T^{2} \)
73 \( 1 + (7.06 + 4.07i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.17 + 4.72i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.7iT - 83T^{2} \)
89 \( 1 + (-10.1 + 5.88i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.01iT - 97T^{2} \)
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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

−10.04487385037002531213095494177, −8.879703080245961134378828163634, −7.50700995907138934706558175330, −7.36887358631373589344170462001, −6.35375435082363172846662268393, −5.50689275831665683818884114813, −4.82664398828016643493720869972, −3.52463562597827592396525798362, −1.91634122930249929079129564419, −0.815995539222114297909015093912, 0.984735621816659473991198731372, 2.64397357403846879263682479836, 4.13165104194504194123938002720, 4.97758509533203623342485148986, 5.62039120291021134275318587417, 6.12298537710903187562404556367, 7.53077335775137699792961477611, 8.299386841153113096178776531266, 9.588613273206719017486222978590, 9.735688221830544223042684816234

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