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Properties

Label 2-1800-360.139-c0-0-5
Degree $2$
Conductor $1800$
Sign $-0.803 + 0.595i$
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  + (0.866 − 0.5i)2-s − i·3-s + (0.499 − 0.866i)4-s + (−0.5 − 0.866i)6-s − 0.999i·8-s − 9-s + (−1 − 1.73i)11-s + (−0.866 − 0.499i)12-s + (−0.5 − 0.866i)16-s + 2i·17-s + (−0.866 + 0.5i)18-s + 19-s + (−1.73 − 0.999i)22-s − 0.999·24-s + i·27-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s − i·3-s + (0.499 − 0.866i)4-s + (−0.5 − 0.866i)6-s − 0.999i·8-s − 9-s + (−1 − 1.73i)11-s + (−0.866 − 0.499i)12-s + (−0.5 − 0.866i)16-s + 2i·17-s + (−0.866 + 0.5i)18-s + 19-s + (−1.73 − 0.999i)22-s − 0.999·24-s + i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.803 + 0.595i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ -0.803 + 0.595i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.688336460\)
\(L(\frac12)\) \(\approx\) \(1.688336460\)
\(L(1)\) 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 \( 1 + iT \)
5 \( 1 \)
good7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - 2iT - T^{2} \)
19 \( 1 - T + T^{2} \)
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 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - iT - T^{2} \)
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 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
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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

−9.021680759950208580206402052421, −8.268573718511528323156575931944, −7.53080945895070580368428882615, −6.52405282222904661379731266311, −5.72472160868746306044532205829, −5.45005675790451733110502725706, −3.90176459506674130362430808153, −3.13239364675335750330635680094, −2.20161079497290581821727772909, −0.971337443057665932736069073532, 2.42726129434054565837119752769, 3.07487805583731034240003614041, 4.28351217985385183458567704774, 4.95082932166739695229391989134, 5.33699363805640430605453411604, 6.51853664307276058990110509528, 7.51026335225660854768938744165, 7.83443294557618095108943511316, 9.236991300132374537259623017989, 9.605271333251628969053296632134

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