Properties

Label 2-2240-40.29-c1-0-9
Degree $2$
Conductor $2240$
Sign $-0.516 - 0.856i$
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  − 0.236·3-s + (1.55 − 1.61i)5-s + i·7-s − 2.94·9-s + 3.36i·11-s − 1.01·13-s + (−0.366 + 0.380i)15-s + 0.988i·17-s + 2.29i·19-s − 0.236i·21-s − 0.806i·23-s + (−0.187 − 4.99i)25-s + 1.40·27-s + 3.17i·29-s + 1.77·31-s + ⋯
L(s)  = 1  − 0.136·3-s + (0.693 − 0.720i)5-s + 0.377i·7-s − 0.981·9-s + 1.01i·11-s − 0.281·13-s + (−0.0946 + 0.0982i)15-s + 0.239i·17-s + 0.526i·19-s − 0.0515i·21-s − 0.168i·23-s + (−0.0374 − 0.999i)25-s + 0.270·27-s + 0.589i·29-s + 0.317·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.516 - 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.516 - 0.856i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.516 - 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8066358152\)
\(L(\frac12)\) \(\approx\) \(0.8066358152\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.55 + 1.61i)T \)
7 \( 1 - iT \)
good3 \( 1 + 0.236T + 3T^{2} \)
11 \( 1 - 3.36iT - 11T^{2} \)
13 \( 1 + 1.01T + 13T^{2} \)
17 \( 1 - 0.988iT - 17T^{2} \)
19 \( 1 - 2.29iT - 19T^{2} \)
23 \( 1 + 0.806iT - 23T^{2} \)
29 \( 1 - 3.17iT - 29T^{2} \)
31 \( 1 - 1.77T + 31T^{2} \)
37 \( 1 + 9.11T + 37T^{2} \)
41 \( 1 + 8.66T + 41T^{2} \)
43 \( 1 + 11.0T + 43T^{2} \)
47 \( 1 - 9.87iT - 47T^{2} \)
53 \( 1 + 4.36T + 53T^{2} \)
59 \( 1 - 2.34iT - 59T^{2} \)
61 \( 1 - 10.2iT - 61T^{2} \)
67 \( 1 - 6.35T + 67T^{2} \)
71 \( 1 - 12.5T + 71T^{2} \)
73 \( 1 + 13.0iT - 73T^{2} \)
79 \( 1 - 7.51T + 79T^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 - 0.147T + 89T^{2} \)
97 \( 1 - 9.75iT - 97T^{2} \)
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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

−9.231519104054121480682275266409, −8.595999413946003078381414895966, −7.945239911603455622448361536588, −6.80862487453465090041837276176, −6.12677347960018554604855430228, −5.19868114382778780509235533107, −4.83181191375597480442385684887, −3.50941168117787393765606624769, −2.38579808187256167114697813720, −1.50341997932227205344299810509, 0.26416342935313920099984678867, 1.89588200972275652488642511362, 2.98685213965485814514147732211, 3.56874979419116426482063113338, 5.06067537251182465517134184126, 5.56623837681578289523333904338, 6.56432675629719082153582796830, 6.94184481629472860752812593103, 8.184005159615061244667339400242, 8.653265009161763210590412323011

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