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Properties

Label 2-1120-35.9-c1-0-45
Degree $2$
Conductor $1120$
Sign $-0.0814 + 0.996i$
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  + (2.96 − 1.71i)3-s + (1.11 − 1.93i)5-s + (−2.45 + 0.978i)7-s + (4.35 − 7.53i)9-s − 7.64i·15-s + (−5.60 + 7.10i)21-s + (7.61 + 4.39i)23-s + (−2.5 − 4.33i)25-s − 19.4i·27-s − 10.7·29-s + (−0.854 + 5.85i)35-s + 8.15·41-s + 6.34i·43-s + (−9.72 − 16.8i)45-s + (0.252 + 0.145i)47-s + ⋯
L(s)  = 1  + (1.71 − 0.987i)3-s + (0.499 − 0.866i)5-s + (−0.929 + 0.369i)7-s + (1.45 − 2.51i)9-s − 1.97i·15-s + (−1.22 + 1.54i)21-s + (1.58 + 0.917i)23-s + (−0.5 − 0.866i)25-s − 3.75i·27-s − 1.99·29-s + (−0.144 + 0.989i)35-s + 1.27·41-s + 0.968i·43-s + (−1.45 − 2.51i)45-s + (0.0368 + 0.0212i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0814 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0814 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $-0.0814 + 0.996i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ -0.0814 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.904944756\)
\(L(\frac12)\) \(\approx\) \(2.904944756\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 + 1.93i)T \)
7 \( 1 + (2.45 - 0.978i)T \)
good3 \( 1 + (-2.96 + 1.71i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.61 - 4.39i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 10.7T + 29T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.15T + 41T^{2} \)
43 \( 1 - 6.34iT - 43T^{2} \)
47 \( 1 + (-0.252 - 0.145i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.109 - 0.190i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.00501 + 0.00289i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 17.4iT - 83T^{2} \)
89 \( 1 + (9.24 - 16.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.209185302841169590060404067854, −9.055710408226355383711513588167, −8.010831608255268832429643986357, −7.30118355891389112818819388422, −6.45391607773283735185581388184, −5.48933570723446394427814629103, −4.01094370897625591611622393192, −3.10738437405720817547154322585, −2.19624788609899764716748656860, −1.10749796527273381639467737211, 2.08600909806357052487195270288, 2.99839709739148115431751049357, 3.57017365177429032496621149298, 4.54221888707160272031164747864, 5.78176735646894608055362162719, 7.08589128302508801391757381144, 7.48005402429524672019494818503, 8.739445982029656730650662366738, 9.263485501268265631084628624065, 9.903893919151560764731936204374

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