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Properties

Label 2-1120-56.19-c1-0-8
Degree $2$
Conductor $1120$
Sign $0.197 - 0.980i$
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  + (1.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + (−0.866 + 2.5i)7-s + (−0.732 − 1.26i)11-s − 4.73·13-s + 1.73i·15-s + (−1.09 + 0.633i)17-s + (7.09 + 4.09i)19-s + (0.866 + 4.5i)21-s + (5.13 + 2.96i)23-s + (−0.499 − 0.866i)25-s + 5.19i·27-s + 10.4i·29-s + (1.09 + 1.90i)31-s + (−2.19 − 1.26i)33-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (−0.223 + 0.387i)5-s + (−0.327 + 0.944i)7-s + (−0.220 − 0.382i)11-s − 1.31·13-s + 0.447i·15-s + (−0.266 + 0.153i)17-s + (1.62 + 0.940i)19-s + (0.188 + 0.981i)21-s + (1.07 + 0.618i)23-s + (−0.0999 − 0.173i)25-s + 0.999i·27-s + 1.94i·29-s + (0.197 + 0.341i)31-s + (−0.382 − 0.220i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.197 - 0.980i$
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: \((2,\ 1120,\ (\ :1/2),\ 0.197 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.575818010\)
\(L(\frac12)\) \(\approx\) \(1.575818010\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 2.5i)T \)
good3 \( 1 + (-1.5 + 0.866i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.732 + 1.26i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.73T + 13T^{2} \)
17 \( 1 + (1.09 - 0.633i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-7.09 - 4.09i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.13 - 2.96i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 10.4iT - 29T^{2} \)
31 \( 1 + (-1.09 - 1.90i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.73 + i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.19iT - 41T^{2} \)
43 \( 1 + 3.92T + 43T^{2} \)
47 \( 1 + (-1.26 + 2.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.169 - 0.0980i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.09 - 2.36i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.13 - 3.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.69 - 9.86i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.26iT - 71T^{2} \)
73 \( 1 + (10.0 - 5.83i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.09 + 2.36i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.6iT - 83T^{2} \)
89 \( 1 + (-12.6 - 7.33i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.46iT - 97T^{2} \)
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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

−9.853540508125586420924454464840, −9.029317544455347705653211718018, −8.445147340892532476182315106925, −7.39826588735854276839113255174, −7.08682813577040244267530944080, −5.66129915109646971856104803332, −5.04931020515767946006300833041, −3.29112845687515044881011406181, −2.91221024659776373386288303872, −1.71897630001949185639619578468, 0.60877347092285853240126160239, 2.52512946950264574630371684254, 3.32474668786633803695511252084, 4.44782387835381760201072896170, 4.96806694017683452803694574897, 6.46020817503341401799832137982, 7.40308697685434963154754645427, 7.917696383157269289882745798542, 9.041230821907177470976938020930, 9.659972148739305428261893377463

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