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Properties

Label 2-2700-540.167-c0-0-1
Degree $2$
Conductor $2700$
Sign $0.835 + 0.550i$
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  + (0.996 + 0.0871i)2-s + (0.422 − 0.906i)3-s + (0.984 + 0.173i)4-s + (0.5 − 0.866i)6-s + (0.560 + 0.392i)7-s + (0.965 + 0.258i)8-s + (−0.642 − 0.766i)9-s + (0.573 − 0.819i)12-s + (0.524 + 0.439i)14-s + (0.939 + 0.342i)16-s + (−0.573 − 0.819i)18-s + (0.592 − 0.342i)21-s + (−0.199 − 0.284i)23-s + (0.642 − 0.766i)24-s + (−0.965 + 0.258i)27-s + (0.483 + 0.483i)28-s + ⋯
L(s)  = 1  + (0.996 + 0.0871i)2-s + (0.422 − 0.906i)3-s + (0.984 + 0.173i)4-s + (0.5 − 0.866i)6-s + (0.560 + 0.392i)7-s + (0.965 + 0.258i)8-s + (−0.642 − 0.766i)9-s + (0.573 − 0.819i)12-s + (0.524 + 0.439i)14-s + (0.939 + 0.342i)16-s + (−0.573 − 0.819i)18-s + (0.592 − 0.342i)21-s + (−0.199 − 0.284i)23-s + (0.642 − 0.766i)24-s + (−0.965 + 0.258i)27-s + (0.483 + 0.483i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.835 + 0.550i$
Analytic conductor: \(1.34747\)
Root analytic conductor: \(1.16080\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :0),\ 0.835 + 0.550i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.757814708\)
\(L(\frac12)\) \(\approx\) \(2.757814708\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.996 - 0.0871i)T \)
3 \( 1 + (-0.422 + 0.906i)T \)
5 \( 1 \)
good7 \( 1 + (-0.560 - 0.392i)T + (0.342 + 0.939i)T^{2} \)
11 \( 1 + (0.766 + 0.642i)T^{2} \)
13 \( 1 + (0.984 - 0.173i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.199 + 0.284i)T + (-0.342 + 0.939i)T^{2} \)
29 \( 1 + (1.50 - 1.26i)T + (0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.939 + 0.342i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T^{2} \)
41 \( 1 + (0.826 - 0.984i)T + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + (0.731 + 1.56i)T + (-0.642 + 0.766i)T^{2} \)
47 \( 1 + (-0.878 + 1.25i)T + (-0.342 - 0.939i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (-1.28 + 0.112i)T + (0.984 - 0.173i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.133 - 1.52i)T + (-0.984 - 0.173i)T^{2} \)
89 \( 1 + (0.642 + 1.11i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.642 + 0.766i)T^{2} \)
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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

−8.548641795500385332345095119113, −8.202677258512160889230585516291, −7.04501743411617800552510362328, −6.92976337982003637841915980402, −5.66350526126018542857815450556, −5.32329189986284556445632331175, −4.08320421530801451316925364150, −3.27432144532511703535870885137, −2.29015137318164710041831128273, −1.54737301774132153295091990903, 1.73006288381997480300862381656, 2.71421113240285438644760109942, 3.70161184413184259490678739798, 4.24929176231564620042439479832, 5.06461624157069535937422452383, 5.71821537405312765734417916364, 6.66200538018671417003450419465, 7.75250440272465322918617875480, 8.054375773608060400033783635337, 9.309296817579175321593277974393

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