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Properties

Label 2-1120-280.179-c0-0-1
Degree $2$
Conductor $1120$
Sign $-0.895 + 0.444i$
Analytic cond. $0.558952$
Root an. cond. $0.747631$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)5-s − 7-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s − 13-s + (−0.5 − 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.5 + 0.866i)35-s + (0.5 + 0.866i)37-s − 41-s + (−0.499 + 0.866i)45-s + (−0.5 − 0.866i)47-s + 49-s + (0.5 − 0.866i)53-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s − 7-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s − 13-s + (−0.5 − 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.5 + 0.866i)35-s + (0.5 + 0.866i)37-s − 41-s + (−0.499 + 0.866i)45-s + (−0.5 − 0.866i)47-s + 49-s + (0.5 − 0.866i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(0.558952\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (879, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :0),\ -0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3464367166\)
\(L(\frac12)\) \(\approx\) \(0.3464367166\)
\(L(1)\) 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 \( 1 + T \)
good3 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.709131831028727635959237003598, −8.882849182052160919112437351062, −8.168754301810118842031985637516, −7.08758870790647450891163429356, −6.46173429197694358847415361560, −5.26396318446279451999506202372, −4.50105680923516349541086778094, −3.44798369180119781821639867289, −2.32529432360541342104995221585, −0.28562293778862235321863141056, 2.37962729814885197046488653328, 3.14245799167679114699953558451, 4.11616225416496281670907335725, 5.48047720458334428457373047470, 6.12500199597531378076261120364, 7.19385487108510001187299243654, 7.81078384865179824066837297089, 8.627004750211032363627211464254, 9.783567556443048594106761603668, 10.36763588970957441546389593178

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