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Properties

Label 2-1800-5.3-c2-0-35
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: $\chi_{1800} (793, \cdot )$
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

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

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