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Properties

Label 2-2700-108.67-c0-0-1
Degree $2$
Conductor $2700$
Sign $-0.286 + 0.957i$
Analytic cond. $1.34747$
Root an. cond. $1.16080$
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  + (−0.984 + 0.173i)2-s + (0.642 − 0.766i)3-s + (0.939 − 0.342i)4-s + (−0.5 + 0.866i)6-s + (0.524 − 1.43i)7-s + (−0.866 + 0.5i)8-s + (−0.173 − 0.984i)9-s + (0.342 − 0.939i)12-s + (−0.266 + 1.50i)14-s + (0.766 − 0.642i)16-s + (0.342 + 0.939i)18-s + (−0.766 − 1.32i)21-s + (−0.642 − 1.76i)23-s + (−0.173 + 0.984i)24-s + (−0.866 − 0.500i)27-s − 1.53i·28-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (0.642 − 0.766i)3-s + (0.939 − 0.342i)4-s + (−0.5 + 0.866i)6-s + (0.524 − 1.43i)7-s + (−0.866 + 0.5i)8-s + (−0.173 − 0.984i)9-s + (0.342 − 0.939i)12-s + (−0.266 + 1.50i)14-s + (0.766 − 0.642i)16-s + (0.342 + 0.939i)18-s + (−0.766 − 1.32i)21-s + (−0.642 − 1.76i)23-s + (−0.173 + 0.984i)24-s + (−0.866 − 0.500i)27-s − 1.53i·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.286 + 0.957i$
Analytic conductor: \(1.34747\)
Root analytic conductor: \(1.16080\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (2551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :0),\ -0.286 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9783097519\)
\(L(\frac12)\) \(\approx\) \(0.9783097519\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 - 0.173i)T \)
3 \( 1 + (-0.642 + 0.766i)T \)
5 \( 1 \)
good7 \( 1 + (-0.524 + 1.43i)T + (-0.766 - 0.642i)T^{2} \)
11 \( 1 + (-0.173 + 0.984i)T^{2} \)
13 \( 1 + (-0.939 - 0.342i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.642 + 1.76i)T + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \)
31 \( 1 + (-0.766 + 0.642i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
43 \( 1 + (-0.642 - 0.766i)T + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.118 + 0.326i)T + (-0.766 - 0.642i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
67 \( 1 + (-0.342 - 0.0603i)T + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (-0.342 + 0.0603i)T + (0.939 - 0.342i)T^{2} \)
89 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.173 - 0.984i)T^{2} \)
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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

−8.668215478462960942155954387969, −7.987292492760681331729078467521, −7.45506660597276357904128138323, −6.78327496076935701114857377226, −6.21443983189667071583706754333, −4.89219198762681392258003728802, −3.81747760709103412527986523361, −2.79181619916161648210137172965, −1.73461319208688640665134456575, −0.808289401177368800397972528983, 1.79392496398567069569763812558, 2.48359538950325078480539953720, 3.36504854255841829321577638191, 4.40859098661331588517012113835, 5.55988879020140216667723760034, 6.04480208281299186987125181893, 7.42472366617588390356882833819, 7.990479821873603614011478759010, 8.594215582176858476247169049888, 9.288153656828913580462220028896

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