Properties

Label 2-1800-8.5-c1-0-39
Degree $2$
Conductor $1800$
Sign $0.917 + 0.398i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.40 − 0.192i)2-s + (1.92 + 0.540i)4-s − 0.0802·7-s + (−2.59 − 1.12i)8-s − 2.41i·11-s + 5.26i·13-s + (0.112 + 0.0154i)14-s + (3.41 + 2.08i)16-s − 0.255·17-s − 6.95i·19-s + (−0.465 + 3.38i)22-s − 1.64·23-s + (1.01 − 7.38i)26-s + (−0.154 − 0.0433i)28-s + 4.51i·29-s + ⋯
L(s)  = 1  + (−0.990 − 0.136i)2-s + (0.962 + 0.270i)4-s − 0.0303·7-s + (−0.917 − 0.398i)8-s − 0.728i·11-s + 1.46i·13-s + (0.0300 + 0.00413i)14-s + (0.854 + 0.520i)16-s − 0.0620·17-s − 1.59i·19-s + (−0.0993 + 0.721i)22-s − 0.343·23-s + (0.199 − 1.44i)26-s + (−0.0292 − 0.00819i)28-s + 0.838i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.917 + 0.398i$
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: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ 0.917 + 0.398i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.038524722\)
\(L(\frac12)\) \(\approx\) \(1.038524722\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.192i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 0.0802T + 7T^{2} \)
11 \( 1 + 2.41iT - 11T^{2} \)
13 \( 1 - 5.26iT - 13T^{2} \)
17 \( 1 + 0.255T + 17T^{2} \)
19 \( 1 + 6.95iT - 19T^{2} \)
23 \( 1 + 1.64T + 23T^{2} \)
29 \( 1 - 4.51iT - 29T^{2} \)
31 \( 1 - 8.29T + 31T^{2} \)
37 \( 1 - 2.67iT - 37T^{2} \)
41 \( 1 - 8.11T + 41T^{2} \)
43 \( 1 + 4.08iT - 43T^{2} \)
47 \( 1 - 5.70T + 47T^{2} \)
53 \( 1 + 11.5iT - 53T^{2} \)
59 \( 1 - 12.6iT - 59T^{2} \)
61 \( 1 + 11.9iT - 61T^{2} \)
67 \( 1 + 7.27iT - 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 - 5.50T + 79T^{2} \)
83 \( 1 - 9.20iT - 83T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 + 8.50T + 97T^{2} \)
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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

−9.207879294246826009784068501772, −8.584268055614635136005511535113, −7.82182983308197088975333097776, −6.70801088800475187313756793340, −6.53989934217533913833350795806, −5.22286475079971363246228640879, −4.11625340654151903152136972298, −2.99546464555679630585753658182, −2.05985667270012139230754413753, −0.73577810483741936086093001696, 0.884012783868078361687189348819, 2.15790180882278036575941617438, 3.13527494556327164187592507861, 4.35785793050621991164474396894, 5.64787770589571723464644157284, 6.12142256911452880723911306712, 7.19799768400644860600469188971, 7.985945029798666842675010511823, 8.271553524849644415520035898285, 9.524795656942592085785676648595

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