Properties

Label 2-2700-4.3-c0-0-1
Degree $2$
Conductor $2700$
Sign $0.866 + 0.5i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 0.999·8-s + (−0.5 − 0.866i)16-s + 17-s + 1.73i·19-s − 1.73i·23-s + 1.73i·31-s + (−0.499 + 0.866i)32-s + (−0.5 − 0.866i)34-s + (1.49 − 0.866i)38-s + (−1.49 + 0.866i)46-s + 49-s + 53-s + 61-s + (1.49 − 0.866i)62-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 0.999·8-s + (−0.5 − 0.866i)16-s + 17-s + 1.73i·19-s − 1.73i·23-s + 1.73i·31-s + (−0.499 + 0.866i)32-s + (−0.5 − 0.866i)34-s + (1.49 − 0.866i)38-s + (−1.49 + 0.866i)46-s + 49-s + 53-s + 61-s + (1.49 − 0.866i)62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.866 + 0.5i$
Analytic conductor: \(1.34747\)
Root analytic conductor: \(1.16080\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (1351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :0),\ 0.866 + 0.5i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9162279792\)
\(L(\frac12)\) \(\approx\) \(0.9162279792\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.872199744927892487379817206518, −8.415922824338715098216996494901, −7.67543391588187900490736525016, −6.85719080956249332986913979381, −5.81855758999976185633811029531, −4.89086501950668731663202192182, −3.94378784983662353489005582332, −3.21791612930898938854884587085, −2.18783905301853992838263396990, −1.09760211608804401461650525018, 0.906711762958472861950307308783, 2.26921688891191251507911889402, 3.58796985750289119547458956224, 4.57330612341103889223627231866, 5.45233726070309140685976454346, 5.96975410646137870802310270271, 7.07813709442973736853300817084, 7.43609004240879064963050263962, 8.269865490636535312864902379023, 9.073899166379738248810567422432

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