↑

Properties

Label 2-1800-40.29-c1-0-35
Degree $2$
Conductor $1800$
Sign $0.714 - 0.700i$
Analytic cond. $14.3730$
Root an. cond. $3.79118$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.576 + 1.29i)2-s + (−1.33 − 1.48i)4-s − 1.97i·7-s + (2.69 − 0.867i)8-s + 1.43i·11-s + 0.241·13-s + (2.55 + 1.13i)14-s + (−0.430 + 3.97i)16-s + 7.38i·17-s − 3.04i·19-s + (−1.84 − 0.824i)22-s + 0.874i·23-s + (−0.139 + 0.311i)26-s + (−2.94 + 2.64i)28-s − 9.07i·29-s + ⋯
L(s)  = 1  + (−0.407 + 0.913i)2-s + (−0.667 − 0.744i)4-s − 0.747i·7-s + (0.951 − 0.306i)8-s + 0.431i·11-s + 0.0669·13-s + (0.682 + 0.304i)14-s + (−0.107 + 0.994i)16-s + 1.79i·17-s − 0.697i·19-s + (−0.393 − 0.175i)22-s + 0.182i·23-s + (−0.0272 + 0.0611i)26-s + (−0.556 + 0.499i)28-s − 1.68i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.714 - 0.700i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ 0.714 - 0.700i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.243060825\)
\(L(\frac12)\) \(\approx\) \(1.243060825\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.576 - 1.29i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 1.97iT - 7T^{2} \)
11 \( 1 - 1.43iT - 11T^{2} \)
13 \( 1 - 0.241T + 13T^{2} \)
17 \( 1 - 7.38iT - 17T^{2} \)
19 \( 1 + 3.04iT - 19T^{2} \)
23 \( 1 - 0.874iT - 23T^{2} \)
29 \( 1 + 9.07iT - 29T^{2} \)
31 \( 1 + 7.44T + 31T^{2} \)
37 \( 1 - 8.81T + 37T^{2} \)
41 \( 1 - 1.91T + 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 - 3.34iT - 47T^{2} \)
53 \( 1 - 9.20T + 53T^{2} \)
59 \( 1 + 6.43iT - 59T^{2} \)
61 \( 1 - 4.57iT - 61T^{2} \)
67 \( 1 - 4.86T + 67T^{2} \)
71 \( 1 - 8.21T + 71T^{2} \)
73 \( 1 - 4.12iT - 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 + 8.08T + 89T^{2} \)
97 \( 1 - 10.6iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.336780036672544657355476796598, −8.436365101007574524256886239603, −7.72420648101045614696007189452, −7.13500988764009386993579700218, −6.20629546146055471712870499911, −5.60859793458397550284453407810, −4.35129718563036017253385714485, −3.93840709408796234853382500520, −2.14630669299834825242352146143, −0.814344833204210115600045507626, 0.827999337902791418997631579266, 2.21431163190392209111310901730, 2.99638165753355487186798741712, 3.95580177175110351237152453906, 5.05668204980208779445880502774, 5.73588076414208839619260997534, 7.06219869051008014694102331870, 7.72931358745077749185179484450, 8.744535499039466489219916163566, 9.158486964571504100781571367815

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