Properties

Label 2-2240-28.27-c1-0-14
Degree $2$
Conductor $2240$
Sign $0.990 - 0.137i$
Analytic cond. $17.8864$
Root an. cond. $4.22924$
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  − 2.76·3-s i·5-s + (−2.62 + 0.363i)7-s + 4.61·9-s − 3.56i·11-s + 0.710i·13-s + 2.76i·15-s + 5.25i·17-s − 7.01·19-s + (7.23 − 1.00i)21-s + 1.77i·23-s − 25-s − 4.46·27-s − 4.08·29-s + 1.18·31-s + ⋯
L(s)  = 1  − 1.59·3-s − 0.447i·5-s + (−0.990 + 0.137i)7-s + 1.53·9-s − 1.07i·11-s + 0.196i·13-s + 0.712i·15-s + 1.27i·17-s − 1.61·19-s + (1.57 − 0.218i)21-s + 0.370i·23-s − 0.200·25-s − 0.859·27-s − 0.757·29-s + 0.213·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.990 - 0.137i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1791, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.990 - 0.137i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4818589240\)
\(L(\frac12)\) \(\approx\) \(0.4818589240\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
7 \( 1 + (2.62 - 0.363i)T \)
good3 \( 1 + 2.76T + 3T^{2} \)
11 \( 1 + 3.56iT - 11T^{2} \)
13 \( 1 - 0.710iT - 13T^{2} \)
17 \( 1 - 5.25iT - 17T^{2} \)
19 \( 1 + 7.01T + 19T^{2} \)
23 \( 1 - 1.77iT - 23T^{2} \)
29 \( 1 + 4.08T + 29T^{2} \)
31 \( 1 - 1.18T + 31T^{2} \)
37 \( 1 + 5.91T + 37T^{2} \)
41 \( 1 + 7.71iT - 41T^{2} \)
43 \( 1 + 6.89iT - 43T^{2} \)
47 \( 1 + 6.90T + 47T^{2} \)
53 \( 1 + 0.578T + 53T^{2} \)
59 \( 1 - 6.12T + 59T^{2} \)
61 \( 1 - 8.34iT - 61T^{2} \)
67 \( 1 - 11.2iT - 67T^{2} \)
71 \( 1 - 15.9iT - 71T^{2} \)
73 \( 1 + 2.37iT - 73T^{2} \)
79 \( 1 + 10.5iT - 79T^{2} \)
83 \( 1 - 16.8T + 83T^{2} \)
89 \( 1 + 14.5iT - 89T^{2} \)
97 \( 1 + 4.55iT - 97T^{2} \)
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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

−8.910469251716138392044648404648, −8.520931640627388633052874406309, −7.20605549735106958796248704074, −6.44507802004840116593382606321, −5.90143665187807053751091929201, −5.39372901670164012579431350156, −4.24564753038364942566093113127, −3.54398593768865927213669524788, −1.93442548721788135424721459875, −0.54064081361871975507078928528, 0.41162471838913462884828126698, 2.04448735574771514409647271598, 3.27507439189943036730506449372, 4.45830852017247938696186160429, 5.01694095559025185111807222715, 6.06811878826347146617929730443, 6.63298381000077134312027953441, 7.04772283814065968208520219807, 8.069827267255807746268457045003, 9.426288712524689791306452113003

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