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Properties

Label 2-1120-56.3-c1-0-19
Degree $2$
Conductor $1120$
Sign $0.833 - 0.552i$
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  + (2.66 + 1.54i)3-s + (−0.5 − 0.866i)5-s + (−2.53 − 0.754i)7-s + (3.24 + 5.61i)9-s + (1.64 − 2.84i)11-s + 6.72·13-s − 3.08i·15-s + (3.32 + 1.91i)17-s + (0.618 − 0.357i)19-s + (−5.60 − 5.91i)21-s + (−1.09 + 0.631i)23-s + (−0.499 + 0.866i)25-s + 10.7i·27-s + 3.04i·29-s + (−0.0335 + 0.0581i)31-s + ⋯
L(s)  = 1  + (1.54 + 0.889i)3-s + (−0.223 − 0.387i)5-s + (−0.958 − 0.285i)7-s + (1.08 + 1.87i)9-s + (0.495 − 0.857i)11-s + 1.86·13-s − 0.795i·15-s + (0.806 + 0.465i)17-s + (0.141 − 0.0819i)19-s + (−1.22 − 1.29i)21-s + (−0.227 + 0.131i)23-s + (−0.0999 + 0.173i)25-s + 2.06i·27-s + 0.565i·29-s + (−0.00602 + 0.0104i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.833 - 0.552i$
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: \((2,\ 1120,\ (\ :1/2),\ 0.833 - 0.552i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.707103666\)
\(L(\frac12)\) \(\approx\) \(2.707103666\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.53 + 0.754i)T \)
good3 \( 1 + (-2.66 - 1.54i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1.64 + 2.84i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.72T + 13T^{2} \)
17 \( 1 + (-3.32 - 1.91i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.618 + 0.357i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.09 - 0.631i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.04iT - 29T^{2} \)
31 \( 1 + (0.0335 - 0.0581i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.498 - 0.287i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.230iT - 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 + (-4.23 - 7.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.16 - 1.25i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.986 + 0.569i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.0888 + 0.153i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.92 + 12.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.0iT - 71T^{2} \)
73 \( 1 + (3.89 + 2.25i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.66 - 5.58i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.24iT - 83T^{2} \)
89 \( 1 + (13.3 - 7.68i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.76iT - 97T^{2} \)
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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

−9.706769619905366300745837506029, −9.006555439794519434168240298789, −8.482734119601105947055068575216, −7.82121969546123463640700653266, −6.57894615635751651528755034352, −5.62870209616687325881632022241, −4.24202961214003842099610908111, −3.53586266268125148752011160343, −3.13981368940621826335288911261, −1.38995859168408126466287896941, 1.28555620004862972399539154850, 2.46702094842994904586533875572, 3.41311174470756479781244265784, 3.95482804441675775544357625669, 5.82467410146009822397579141027, 6.73501958950824127485907242170, 7.21267055987892286770368035867, 8.249235628333356997221480771183, 8.746504294793961999137377888644, 9.620570321121755785822132702364

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