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Properties

Label 2-1120-56.19-c1-0-14
Degree $2$
Conductor $1120$
Sign $0.910 - 0.414i$
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  + (0.908 − 0.524i)3-s + (0.5 − 0.866i)5-s + (2.14 + 1.54i)7-s + (−0.949 + 1.64i)9-s + (1.17 + 2.03i)11-s + 1.21·13-s − 1.04i·15-s + (−4.23 + 2.44i)17-s + (2.21 + 1.27i)19-s + (2.76 + 0.276i)21-s + (7.59 + 4.38i)23-s + (−0.499 − 0.866i)25-s + 5.13i·27-s − 5.21i·29-s + (1.68 + 2.92i)31-s + ⋯
L(s)  = 1  + (0.524 − 0.302i)3-s + (0.223 − 0.387i)5-s + (0.811 + 0.583i)7-s + (−0.316 + 0.548i)9-s + (0.354 + 0.614i)11-s + 0.336·13-s − 0.270i·15-s + (−1.02 + 0.593i)17-s + (0.507 + 0.292i)19-s + (0.602 + 0.0603i)21-s + (1.58 + 0.913i)23-s + (−0.0999 − 0.173i)25-s + 0.989i·27-s − 0.968i·29-s + (0.303 + 0.525i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.910 - 0.414i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ 0.910 - 0.414i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.180198730\)
\(L(\frac12)\) \(\approx\) \(2.180198730\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.14 - 1.54i)T \)
good3 \( 1 + (-0.908 + 0.524i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1.17 - 2.03i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.21T + 13T^{2} \)
17 \( 1 + (4.23 - 2.44i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.21 - 1.27i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.59 - 4.38i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.21iT - 29T^{2} \)
31 \( 1 + (-1.68 - 2.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.16 + 3.55i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.18iT - 41T^{2} \)
43 \( 1 + 11.6T + 43T^{2} \)
47 \( 1 + (-5.01 + 8.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.03 + 4.05i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.90 + 5.72i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.560 - 0.971i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.386 + 0.670i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.53iT - 71T^{2} \)
73 \( 1 + (-11.1 + 6.43i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.34 + 0.776i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.47iT - 83T^{2} \)
89 \( 1 + (9.58 + 5.53i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.3iT - 97T^{2} \)
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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

−9.731292123552523609145728251064, −8.766214462537269513537280704155, −8.488833189680484035831477081032, −7.51532718953358536457701344745, −6.65762358533386422959537078121, −5.42092586789207432335198968037, −4.89428504818565154325964857577, −3.62469999702090549662912190170, −2.30201614550772081468781471477, −1.54904827907694485829782391712, 1.00325642145695714580711293890, 2.59237466303009998958957913806, 3.47112511896335209695109932607, 4.47047737869470503773025788664, 5.41806607301001073170762090727, 6.65173642231327604187098170027, 7.13076829321050834973265312919, 8.490846406235063507005266300756, 8.767158302507101279556160993450, 9.696158632469993110119542800986

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