Properties

Label 2-1120-56.27-c1-0-7
Degree $2$
Conductor $1120$
Sign $-0.504 - 0.863i$
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  + 2.21i·3-s + 5-s + (−1.20 − 2.35i)7-s − 1.92·9-s + 3.88·11-s − 5.67·13-s + 2.21i·15-s + 5.63i·17-s + 1.31i·19-s + (5.22 − 2.68i)21-s + 7.37i·23-s + 25-s + 2.38i·27-s + 9.07i·29-s + 2.23·31-s + ⋯
L(s)  = 1  + 1.28i·3-s + 0.447·5-s + (−0.457 − 0.889i)7-s − 0.641·9-s + 1.17·11-s − 1.57·13-s + 0.572i·15-s + 1.36i·17-s + 0.300i·19-s + (1.13 − 0.585i)21-s + 1.53i·23-s + 0.200·25-s + 0.459i·27-s + 1.68i·29-s + 0.400·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $-0.504 - 0.863i$
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: \((2,\ 1120,\ (\ :1/2),\ -0.504 - 0.863i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.452417791\)
\(L(\frac12)\) \(\approx\) \(1.452417791\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.20 + 2.35i)T \)
good3 \( 1 - 2.21iT - 3T^{2} \)
11 \( 1 - 3.88T + 11T^{2} \)
13 \( 1 + 5.67T + 13T^{2} \)
17 \( 1 - 5.63iT - 17T^{2} \)
19 \( 1 - 1.31iT - 19T^{2} \)
23 \( 1 - 7.37iT - 23T^{2} \)
29 \( 1 - 9.07iT - 29T^{2} \)
31 \( 1 - 2.23T + 31T^{2} \)
37 \( 1 + 6.98iT - 37T^{2} \)
41 \( 1 - 7.47iT - 41T^{2} \)
43 \( 1 - 1.46T + 43T^{2} \)
47 \( 1 + 0.567T + 47T^{2} \)
53 \( 1 + 0.100iT - 53T^{2} \)
59 \( 1 + 2.93iT - 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 + 5.54T + 67T^{2} \)
71 \( 1 - 2.42iT - 71T^{2} \)
73 \( 1 + 6.08iT - 73T^{2} \)
79 \( 1 + 2.83iT - 79T^{2} \)
83 \( 1 - 2.52iT - 83T^{2} \)
89 \( 1 + 10.9iT - 89T^{2} \)
97 \( 1 + 9.93iT - 97T^{2} \)
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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

−9.957724322205110308007666758312, −9.575562115019866585116788846836, −8.808170379612836949021588763881, −7.53219927950420473177794288905, −6.78060947961213427456850076986, −5.73907869234775131054124996537, −4.78379687035194463605901582788, −3.96091181001566835037586608868, −3.27683141211025921504582063759, −1.57539777070236663583569894144, 0.64496432463255853695470338133, 2.22260199789315473179877439408, 2.68456960736280826968484437464, 4.43260170479840404164654942735, 5.45735874322511628775584529836, 6.54218232507875070350261199393, 6.80304992420518118456440892448, 7.78851411323115534116817775293, 8.752206709935286666497747616048, 9.521542628702478206102157598240

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