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Properties

Label 2-2280-2280.1139-c0-0-18
Degree $2$
Conductor $2280$
Sign $-1$
Analytic cond. $1.13786$
Root an. cond. $1.06670$
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  − i·2-s − i·3-s − 4-s + 5-s − 6-s + i·8-s − 9-s − i·10-s + i·12-s − 2i·13-s − i·15-s + 16-s + i·18-s − 19-s − 20-s + ⋯
L(s)  = 1  − i·2-s − i·3-s − 4-s + 5-s − 6-s + i·8-s − 9-s − i·10-s + i·12-s − 2i·13-s − i·15-s + 16-s + i·18-s − 19-s − 20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-1$
Analytic conductor: \(1.13786\)
Root analytic conductor: \(1.06670\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2280,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.093469538\)
\(L(\frac12)\) \(\approx\) \(1.093469538\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + iT \)
5 \( 1 - T \)
19 \( 1 + T \)
good7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 2iT - T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + 2iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
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 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
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

−8.858508389841948373469563998582, −8.185629327418117794144878179685, −7.45720791345826000113438576396, −6.26285437141765876776403317826, −5.66036763978492244613308716832, −4.96374147583365659356249308784, −3.54119215764471194177932792966, −2.65739763368455396958493654866, −1.95740304706196417965702734672, −0.75667258904789512891001073495, 1.83677336623423808097839471787, 3.21260068002483244148863769311, 4.41884089222829078716743584806, 4.70034483020557050021506725365, 5.77709219881919021514761229297, 6.41119860895608522816879231781, 6.99149235018699872112973281628, 8.365430018495659700704419354828, 8.769929631455492155954184338548, 9.606035818896037492644601741008

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