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Properties

Label 2-2700-36.7-c0-0-1
Degree $2$
Conductor $2700$
Sign $-0.173 + 0.984i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)23-s − 0.999i·28-s + (0.5 − 0.866i)29-s + (0.866 − 0.499i)32-s + (−0.5 − 0.866i)41-s + (−1.73 − i)43-s − 0.999·46-s + (0.866 + 0.5i)47-s + (−0.5 + 0.866i)56-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)23-s − 0.999i·28-s + (0.5 − 0.866i)29-s + (0.866 − 0.499i)32-s + (−0.5 − 0.866i)41-s + (−1.73 − i)43-s − 0.999·46-s + (0.866 + 0.5i)47-s + (−0.5 + 0.866i)56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(1.34747\)
Root analytic conductor: \(1.16080\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :0),\ -0.173 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6117864706\)
\(L(\frac12)\) \(\approx\) \(0.6117864706\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−8.791044579781959107722314651783, −8.301573672966944432143518512501, −7.23406170739066301162290690774, −6.84216557926792527555072571925, −5.97069707077278172145685403557, −4.71331755537479533627062239772, −3.71583700336393557688251818921, −3.03371683191849019474949865838, −1.97204088140060324704623947880, −0.56670579021120350992513561776, 1.26358612528574433633198875633, 2.54675585906979557940636393260, 3.39485618204379045521709257352, 4.82817639706075395559889072690, 5.53832768037043317014931884735, 6.42687884755821449236328197771, 6.89164607494742165251858934610, 7.76645004164115085264619163583, 8.609175786171695801163369586995, 9.109526165950445312348622398351

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