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Properties

Label 2-2240-280.69-c0-0-11
Degree $2$
Conductor $2240$
Sign $-0.965 - 0.258i$
Analytic cond. $1.11790$
Root an. cond. $1.05731$
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  − 1.73i·3-s − i·5-s − 7-s − 1.99·9-s − i·11-s − i·13-s − 1.73·15-s + 1.73·17-s + 1.73i·21-s − 25-s + 1.73i·27-s + 1.73i·29-s − 1.73·33-s + i·35-s − 1.73·39-s + ⋯
L(s)  = 1  − 1.73i·3-s − i·5-s − 7-s − 1.99·9-s − i·11-s − i·13-s − 1.73·15-s + 1.73·17-s + 1.73i·21-s − 25-s + 1.73i·27-s + 1.73i·29-s − 1.73·33-s + i·35-s − 1.73·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.965 - 0.258i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :0),\ -0.965 - 0.258i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9055276035\)
\(L(\frac12)\) \(\approx\) \(0.9055276035\)
\(L(1)\) 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 \( 1 + iT \)
7 \( 1 + T \)
good3 \( 1 + 1.73iT - T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 - 1.73T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 1.73iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 1.73T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 1.73T + T^{2} \)
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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

−8.526486990587713156434024728939, −8.033577771725606278311762997066, −7.34270029199508713450591948855, −6.46530551324197007288941980661, −5.72234754024783235099751691898, −5.26980460195156881614733048304, −3.50110640870479773800511543217, −2.92207768720513398570167301582, −1.47346217228503113162456426327, −0.64859759517093075651645803604, 2.33350526209168996816095608825, 3.30695204261807529338828167017, 3.86298780968657571311390244287, 4.67769878768498620042345203235, 5.69415921864783732190158249897, 6.39080165775496994034586347093, 7.28007798869464178411395341572, 8.176315676170151173289727831185, 9.456546576808327563586073728101, 9.625989720275387775725160556598

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