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Properties

Label 2-1800-5.2-c2-0-9
Degree $2$
Conductor $1800$
Sign $0.130 - 0.991i$
Analytic cond. $49.0464$
Root an. cond. $7.00331$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.775 + 0.775i)7-s − 2.89·11-s + (5.87 − 5.87i)13-s + (−4.44 − 4.44i)17-s + 0.101i·19-s + (−25.3 + 25.3i)23-s + 32.2i·29-s − 3.69·31-s + (42.6 + 42.6i)37-s + 12.8·41-s + (49.2 − 49.2i)43-s + (−2.85 − 2.85i)47-s − 47.7i·49-s + (13.1 − 13.1i)53-s + 76.3i·59-s + ⋯
L(s)  = 1  + (0.110 + 0.110i)7-s − 0.263·11-s + (0.452 − 0.452i)13-s + (−0.261 − 0.261i)17-s + 0.00531i·19-s + (−1.10 + 1.10i)23-s + 1.11i·29-s − 0.119·31-s + (1.15 + 1.15i)37-s + 0.314·41-s + (1.14 − 1.14i)43-s + (−0.0607 − 0.0607i)47-s − 0.975i·49-s + (0.248 − 0.248i)53-s + 1.29i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.130 - 0.991i$
Analytic conductor: \(49.0464\)
Root analytic conductor: \(7.00331\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1),\ 0.130 - 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.456799215\)
\(L(\frac12)\) \(\approx\) \(1.456799215\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.775 - 0.775i)T + 49iT^{2} \)
11 \( 1 + 2.89T + 121T^{2} \)
13 \( 1 + (-5.87 + 5.87i)T - 169iT^{2} \)
17 \( 1 + (4.44 + 4.44i)T + 289iT^{2} \)
19 \( 1 - 0.101iT - 361T^{2} \)
23 \( 1 + (25.3 - 25.3i)T - 529iT^{2} \)
29 \( 1 - 32.2iT - 841T^{2} \)
31 \( 1 + 3.69T + 961T^{2} \)
37 \( 1 + (-42.6 - 42.6i)T + 1.36e3iT^{2} \)
41 \( 1 - 12.8T + 1.68e3T^{2} \)
43 \( 1 + (-49.2 + 49.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (2.85 + 2.85i)T + 2.20e3iT^{2} \)
53 \( 1 + (-13.1 + 13.1i)T - 2.80e3iT^{2} \)
59 \( 1 - 76.3iT - 3.48e3T^{2} \)
61 \( 1 + 103.T + 3.72e3T^{2} \)
67 \( 1 + (-47.6 - 47.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 29.7T + 5.04e3T^{2} \)
73 \( 1 + (-3.50 + 3.50i)T - 5.32e3iT^{2} \)
79 \( 1 - 87.7iT - 6.24e3T^{2} \)
83 \( 1 + (81.7 - 81.7i)T - 6.88e3iT^{2} \)
89 \( 1 - 96.5iT - 7.92e3T^{2} \)
97 \( 1 + (-54.2 - 54.2i)T + 9.40e3iT^{2} \)
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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

−9.248901652510923336147919855271, −8.460519657389497820239756935211, −7.73709697951279962685283634901, −6.96332391059776196753766429049, −5.94489633771219477487223352480, −5.33612749955117143493895916255, −4.27878885404024517968067842276, −3.36723080668149735996675848332, −2.31893073668369253894680283254, −1.09771582848561175156042867281, 0.41535985178890187489006940072, 1.83236080145424271601819528335, 2.80589021546897552873257724815, 4.09054048444974370924700876625, 4.57552743998593411861341398884, 5.96060018043646280660579767406, 6.27454299337675060791667412915, 7.51471928373864288097616847602, 8.025173303916652066245691655564, 8.956293966924682042569459768558

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