Properties

Label 2-1800-8.5-c1-0-86
Degree $2$
Conductor $1800$
Sign $-i$
Analytic cond. $14.3730$
Root an. cond. $3.79118$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 2.00·4-s − 2·7-s + 2.82i·8-s − 5.65i·11-s + 2.82i·14-s + 4.00·16-s − 8.00·22-s + 4.00·28-s + 2.82i·29-s − 10·31-s − 5.65i·32-s + 11.3i·44-s − 3·49-s + 14.1i·53-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s − 0.755·7-s + 1.00i·8-s − 1.70i·11-s + 0.755i·14-s + 1.00·16-s − 1.70·22-s + 0.755·28-s + 0.525i·29-s − 1.79·31-s − 1.00i·32-s + 1.70i·44-s − 0.428·49-s + 1.94i·53-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + 5.65iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 14.1iT - 53T^{2} \)
59 \( 1 - 11.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 5.65iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−8.947174569086627530166685160944, −8.186389872541100225267746582602, −7.16980034059035337689784753713, −5.97841109450943491439673499190, −5.48695190598253148257586371512, −4.22532595064622352166774962568, −3.38481122187681106371870946495, −2.76014854368415329163206817809, −1.30825775781324601032672761017, 0, 1.87994880808488266813241989632, 3.36275080870364268618296400089, 4.27738163237587234390225253332, 5.08098196072103224937510906616, 5.95937510888850869399184321810, 6.87201030730388695291060964067, 7.27555897520815826322214297244, 8.159503348502677388655341050829, 9.097278333646016043644896632751

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