Properties

Label 2-1120-56.27-c1-0-12
Degree $2$
Conductor $1120$
Sign $-0.385 - 0.922i$
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  + 3.19i·3-s + 5-s + (2.59 − 0.525i)7-s − 7.23·9-s + 3.34·11-s + 3.90·13-s + 3.19i·15-s − 2.92i·17-s + 6.33i·19-s + (1.68 + 8.29i)21-s + 3.44i·23-s + 25-s − 13.5i·27-s + 2.68i·29-s + 2.52·31-s + ⋯
L(s)  = 1  + 1.84i·3-s + 0.447·5-s + (0.980 − 0.198i)7-s − 2.41·9-s + 1.00·11-s + 1.08·13-s + 0.826i·15-s − 0.710i·17-s + 1.45i·19-s + (0.366 + 1.81i)21-s + 0.718i·23-s + 0.200·25-s − 2.61i·27-s + 0.497i·29-s + 0.453·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $-0.385 - 0.922i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ -0.385 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.040765845\)
\(L(\frac12)\) \(\approx\) \(2.040765845\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 + (-2.59 + 0.525i)T \)
good3 \( 1 - 3.19iT - 3T^{2} \)
11 \( 1 - 3.34T + 11T^{2} \)
13 \( 1 - 3.90T + 13T^{2} \)
17 \( 1 + 2.92iT - 17T^{2} \)
19 \( 1 - 6.33iT - 19T^{2} \)
23 \( 1 - 3.44iT - 23T^{2} \)
29 \( 1 - 2.68iT - 29T^{2} \)
31 \( 1 - 2.52T + 31T^{2} \)
37 \( 1 - 4.70iT - 37T^{2} \)
41 \( 1 + 5.59iT - 41T^{2} \)
43 \( 1 + 8.62T + 43T^{2} \)
47 \( 1 - 0.506T + 47T^{2} \)
53 \( 1 + 11.4iT - 53T^{2} \)
59 \( 1 + 0.802iT - 59T^{2} \)
61 \( 1 + 7.97T + 61T^{2} \)
67 \( 1 + 6.37T + 67T^{2} \)
71 \( 1 - 1.11iT - 71T^{2} \)
73 \( 1 + 5.91iT - 73T^{2} \)
79 \( 1 + 10.5iT - 79T^{2} \)
83 \( 1 - 4.29iT - 83T^{2} \)
89 \( 1 + 2.00iT - 89T^{2} \)
97 \( 1 - 6.06iT - 97T^{2} \)
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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

−10.15704875139814901539492492007, −9.316741808048547752362352662101, −8.723565274556107539061743545944, −7.945066813356735996803084220458, −6.48119147758880497828185622636, −5.57279283913996351831672303254, −4.87049461343520043699650579705, −3.95190413372822636407151323248, −3.31102451629605035860800932666, −1.58500862881503983071344195183, 1.04183101989741240900579291263, 1.79653246344552126091756004930, 2.84414173718037702448099003207, 4.39961171340731282755769732759, 5.67894687628465972207337628389, 6.35531815090584674825849497741, 6.95719504039669181723142928993, 7.959155993052166818189819239590, 8.578162834950648889085823245040, 9.173468364114977577579634496425

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