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Properties

Label 2-1120-56.19-c1-0-23
Degree $2$
Conductor $1120$
Sign $-0.117 + 0.993i$
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  + (−0.725 + 0.418i)3-s + (−0.5 + 0.866i)5-s + (2.36 − 1.17i)7-s + (−1.14 + 1.99i)9-s + (−2.98 − 5.17i)11-s − 2.87·13-s − 0.837i·15-s + (2.07 − 1.19i)17-s + (4.05 + 2.34i)19-s + (−1.22 + 1.84i)21-s + (−5.81 − 3.35i)23-s + (−0.499 − 0.866i)25-s − 4.43i·27-s + 1.31i·29-s + (−4.34 − 7.53i)31-s + ⋯
L(s)  = 1  + (−0.418 + 0.241i)3-s + (−0.223 + 0.387i)5-s + (0.895 − 0.445i)7-s + (−0.382 + 0.663i)9-s + (−0.901 − 1.56i)11-s − 0.797·13-s − 0.216i·15-s + (0.503 − 0.290i)17-s + (0.930 + 0.537i)19-s + (−0.267 + 0.403i)21-s + (−1.21 − 0.700i)23-s + (−0.0999 − 0.173i)25-s − 0.854i·27-s + 0.243i·29-s + (−0.780 − 1.35i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $-0.117 + 0.993i$
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: \((2,\ 1120,\ (\ :1/2),\ -0.117 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7755460925\)
\(L(\frac12)\) \(\approx\) \(0.7755460925\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.36 + 1.17i)T \)
good3 \( 1 + (0.725 - 0.418i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (2.98 + 5.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.87T + 13T^{2} \)
17 \( 1 + (-2.07 + 1.19i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.05 - 2.34i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.81 + 3.35i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.31iT - 29T^{2} \)
31 \( 1 + (4.34 + 7.53i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.26 + 1.88i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.79iT - 41T^{2} \)
43 \( 1 - 6.73T + 43T^{2} \)
47 \( 1 + (0.874 - 1.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.0994 - 0.0574i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.61 - 1.50i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.40 + 7.62i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.83 - 4.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.06iT - 71T^{2} \)
73 \( 1 + (-8.69 + 5.01i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (11.9 + 6.91i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 17.4iT - 83T^{2} \)
89 \( 1 + (11.0 + 6.38i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.76iT - 97T^{2} \)
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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

−9.893828804266124374477865924359, −8.583726575290422081290200152572, −7.83703507506910098781810442114, −7.44375643933603626803710458444, −5.90025081317892411725870494238, −5.45392313430601924131770179056, −4.47528587030362438794978363681, −3.33220087295950559253356174065, −2.21978923699542764777236802151, −0.35865335041516767055203000510, 1.46751600614405678945726509313, 2.63410112168624972627111257555, 4.06251172436098062145357044656, 5.22080742272690288442770055345, 5.42133174199956922050065952507, 6.88656616130859441617633754290, 7.56687777926271888904257477423, 8.275374834804602072488692502864, 9.345802641724826784251324331135, 9.930768150810325231148816606168

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