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Properties

Label 2-1800-5.2-c2-0-39
Degree $2$
Conductor $1800$
Sign $-0.973 + 0.229i$
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  + (−0.298 − 0.298i)7-s − 11.4·11-s + (1.59 − 1.59i)13-s + (10.4 + 10.4i)17-s + 2.80i·19-s + (19.1 − 19.1i)23-s − 5.61i·29-s − 15.4·31-s + (40.6 + 40.6i)37-s − 70.6·41-s + (−20.2 + 20.2i)43-s + (−42.5 − 42.5i)47-s − 48.8i·49-s + (−3 + 3i)53-s − 66.8i·59-s + ⋯
L(s)  = 1  + (−0.0426 − 0.0426i)7-s − 1.03·11-s + (0.122 − 0.122i)13-s + (0.611 + 0.611i)17-s + 0.147i·19-s + (0.830 − 0.830i)23-s − 0.193i·29-s − 0.496·31-s + (1.09 + 1.09i)37-s − 1.72·41-s + (−0.472 + 0.472i)43-s + (−0.904 − 0.904i)47-s − 0.996i·49-s + (−0.0566 + 0.0566i)53-s − 1.13i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.973 + 0.229i$
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: \((2,\ 1800,\ (\ :1),\ -0.973 + 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2089050879\)
\(L(\frac12)\) \(\approx\) \(0.2089050879\)
\(L(2)\) 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 \( 1 + (0.298 + 0.298i)T + 49iT^{2} \)
11 \( 1 + 11.4T + 121T^{2} \)
13 \( 1 + (-1.59 + 1.59i)T - 169iT^{2} \)
17 \( 1 + (-10.4 - 10.4i)T + 289iT^{2} \)
19 \( 1 - 2.80iT - 361T^{2} \)
23 \( 1 + (-19.1 + 19.1i)T - 529iT^{2} \)
29 \( 1 + 5.61iT - 841T^{2} \)
31 \( 1 + 15.4T + 961T^{2} \)
37 \( 1 + (-40.6 - 40.6i)T + 1.36e3iT^{2} \)
41 \( 1 + 70.6T + 1.68e3T^{2} \)
43 \( 1 + (20.2 - 20.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (42.5 + 42.5i)T + 2.20e3iT^{2} \)
53 \( 1 + (3 - 3i)T - 2.80e3iT^{2} \)
59 \( 1 + 66.8iT - 3.48e3T^{2} \)
61 \( 1 + 44.5T + 3.72e3T^{2} \)
67 \( 1 + (28.2 + 28.2i)T + 4.48e3iT^{2} \)
71 \( 1 + 32.5T + 5.04e3T^{2} \)
73 \( 1 + (62.6 - 62.6i)T - 5.32e3iT^{2} \)
79 \( 1 - 118. iT - 6.24e3T^{2} \)
83 \( 1 + (11.2 - 11.2i)T - 6.88e3iT^{2} \)
89 \( 1 - 11.2iT - 7.92e3T^{2} \)
97 \( 1 + (49 + 49i)T + 9.40e3iT^{2} \)
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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

−8.442304821193598823590371186850, −8.164581204692672901829270602696, −7.12902477756198360911336085545, −6.35447320627881713953004001770, −5.40068031861376572940143466461, −4.72862588607603887959526015222, −3.55243480390687330185677138275, −2.72192175529928343792554215686, −1.48455050810587335893577785462, −0.05424790399642572758179719974, 1.37822461525569018074771707318, 2.67354321871273166357406565722, 3.44189286373947109683212035060, 4.68164796391330076772356699048, 5.36569402101479854844374968587, 6.18743582859492781931150503333, 7.32266558292740569696132102621, 7.69181382778084544900462351575, 8.752948461438179427575914693156, 9.405305148389946954454979641424

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