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Properties

Label 2-2700-4.3-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(1.197148042\)
\(L(\frac12)\) \(\approx\) \(1.197148042\)
\(L(1)\) 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

−9.104078691853285542413048984545, −8.461710531817047595652523021846, −7.68852337728829215182320828084, −7.06587418240520571348815388056, −6.21157062355082995552424960052, −5.56489199863797369699894329831, −4.76265738435860409091044177765, −3.83714398275459320413460465217, −3.15905939769628277495143312069, −1.74790821119038597684203659230, 0.64314962498935718403937317123, 2.24302317789829133255427653163, 2.72792543766052268805591123540, 4.03379524078137609651728418066, 4.55904004132696618588732009723, 5.39466242148143039027461352434, 6.38898256498316002628306783656, 6.92397778096216409542147957616, 8.178365256793128375967891801974, 8.917007209856581916383616280775

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