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Properties

Label 2-440-5.4-c1-0-8
Degree $2$
Conductor $440$
Sign $-0.0536 + 0.998i$
Analytic cond. $3.51341$
Root an. cond. $1.87441$
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  − 0.655i·3-s + (−2.23 − 0.119i)5-s − 0.415i·7-s + 2.56·9-s + 11-s − 4i·13-s + (−0.0786 + 1.46i)15-s − 6.51i·17-s − 5.20·19-s − 0.272·21-s − 8.54i·23-s + (4.97 + 0.535i)25-s − 3.65i·27-s − 0.895·29-s − 6.73·31-s + ⋯
L(s)  = 1  − 0.378i·3-s + (−0.998 − 0.0536i)5-s − 0.157i·7-s + 0.856·9-s + 0.301·11-s − 1.10i·13-s + (−0.0203 + 0.378i)15-s − 1.58i·17-s − 1.19·19-s − 0.0595·21-s − 1.78i·23-s + (0.994 + 0.107i)25-s − 0.702i·27-s − 0.166·29-s − 1.21·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0536 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0536 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $-0.0536 + 0.998i$
Analytic conductor: \(3.51341\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{440} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :1/2),\ -0.0536 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.736865 - 0.777522i\)
\(L(\frac12)\) \(\approx\) \(0.736865 - 0.777522i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.119i)T \)
11 \( 1 - T \)
good3 \( 1 + 0.655iT - 3T^{2} \)
7 \( 1 + 0.415iT - 7T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 6.51iT - 17T^{2} \)
19 \( 1 + 5.20T + 19T^{2} \)
23 \( 1 + 8.54iT - 23T^{2} \)
29 \( 1 + 0.895T + 29T^{2} \)
31 \( 1 + 6.73T + 31T^{2} \)
37 \( 1 - 8.96iT - 37T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
43 \( 1 - 4.78iT - 43T^{2} \)
47 \( 1 - 5.61iT - 47T^{2} \)
53 \( 1 + 10.0iT - 53T^{2} \)
59 \( 1 - 1.63T + 59T^{2} \)
61 \( 1 - 7.10T + 61T^{2} \)
67 \( 1 - 10.6iT - 67T^{2} \)
71 \( 1 + 6.19T + 71T^{2} \)
73 \( 1 + 3.16iT - 73T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 + 16.2iT - 83T^{2} \)
89 \( 1 + 9.56T + 89T^{2} \)
97 \( 1 - 0.591iT - 97T^{2} \)
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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

−10.92448544767714670639191824660, −10.13191538622934850354887719574, −8.949499323788675653568836215114, −8.008468336965559721265162475751, −7.26468473010003212192284123298, −6.43602586885324991492514379280, −4.90416015728241865315916024226, −4.04639869856319331383710435753, −2.67118591724459015046118865110, −0.70845767846003614932981050989, 1.82274532264421898656091915310, 3.97324939615025537803168580871, 4.01735024509585417568340446694, 5.63537858228307536577316149651, 6.87187044376768235780800743523, 7.61901320572662948947084508882, 8.756148642497034167306146836656, 9.446228420935639624955594782389, 10.68853315739640271788519588386, 11.16019838304196693523627526717

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