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Properties

Label 2-1800-72.43-c0-0-2
Degree $2$
Conductor $1800$
Sign $0.766 + 0.642i$
Analytic cond. $0.898317$
Root an. cond. $0.947795$
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 + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 0.999·6-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.5 − 0.866i)11-s + (−0.499 + 0.866i)12-s + (−0.5 + 0.866i)16-s − 17-s + (−0.499 − 0.866i)18-s + 2·19-s + (0.499 + 0.866i)22-s + (−0.499 − 0.866i)24-s + 0.999·27-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 0.999·6-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.5 − 0.866i)11-s + (−0.499 + 0.866i)12-s + (−0.5 + 0.866i)16-s − 17-s + (−0.499 − 0.866i)18-s + 2·19-s + (0.499 + 0.866i)22-s + (−0.499 − 0.866i)24-s + 0.999·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6751098931\)
\(L(\frac12)\) \(\approx\) \(0.6751098931\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
good7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 - 2T + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 2T + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.052828197951982238414009906431, −8.564377193879377025029768840084, −7.60397149181399659920888277447, −7.06814921452909538136307485302, −6.30639808719247771770436192254, −5.57494390043287875995783801860, −4.89899552314441338861927351671, −3.51672202746148954233044226584, −1.97074396961938038664392526891, −0.74476094257423663235645162715, 1.28771373136281373026529088091, 2.73465657168339997348935764546, 3.64547040969726651269707417843, 4.53848517681511873929800462976, 5.14074326690164370213115559887, 6.40933717768054318486031319926, 7.28232565099773069713868721022, 8.224625611435152696121861791144, 9.186739840430260288702629612261, 9.640438578146015356095088056803

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