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Properties

Label 2-1120-56.19-c1-0-4
Degree $2$
Conductor $1120$
Sign $-0.636 - 0.770i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.636 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $-0.636 - 0.770i$
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: \((2,\ 1120,\ (\ :1/2),\ -0.636 - 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6839637306\)
\(L(\frac12)\) \(\approx\) \(0.6839637306\)
\(L(\frac{3}{2})\) 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

−10.04811694644921518324464546534, −9.406424037289407722485814585262, −8.292484209664220001195379515633, −7.74108006646797813595384779428, −6.78706992512139112926832561232, −6.22813243048212638157452421373, −4.80695246738346971657608799717, −3.87880583486717701805611347833, −2.87837030595259803294166133316, −1.83104494035008732155098231750, 0.25993433456892097304707534331, 2.24385094355989960126973784163, 3.31561026032117367734808257523, 4.02915944696320574958497991695, 5.29270552838411665160785702094, 6.14893651227347950534601001866, 6.97512914070917989151899139824, 8.181461386050062470904753548938, 8.756712978140324629879120555559, 9.577071315332387809119373463324

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