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Properties

Label 2-1800-5.2-c2-0-22
Degree $2$
Conductor $1800$
Sign $0.945 + 0.326i$
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  + (−0.325 − 0.325i)7-s + 11.7·11-s + (−3.67 + 3.67i)13-s + (11.3 + 11.3i)17-s − 30.3i·19-s + (9.55 − 9.55i)23-s + 15.3i·29-s − 21.2·31-s + (−3.10 − 3.10i)37-s − 18.2·41-s + (−19.4 + 19.4i)43-s + (27.7 + 27.7i)47-s − 48.7i·49-s + (56.8 − 56.8i)53-s + 82i·59-s + ⋯
L(s)  = 1  + (−0.0465 − 0.0465i)7-s + 1.07·11-s + (−0.282 + 0.282i)13-s + (0.667 + 0.667i)17-s − 1.59i·19-s + (0.415 − 0.415i)23-s + 0.530i·29-s − 0.683·31-s + (−0.0838 − 0.0838i)37-s − 0.443·41-s + (−0.451 + 0.451i)43-s + (0.590 + 0.590i)47-s − 0.995i·49-s + (1.07 − 1.07i)53-s + 1.38i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.945 + 0.326i$
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: \((2,\ 1800,\ (\ :1),\ 0.945 + 0.326i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.122065800\)
\(L(\frac12)\) \(\approx\) \(2.122065800\)
\(L(2)\) 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 \( 1 + (0.325 + 0.325i)T + 49iT^{2} \)
11 \( 1 - 11.7T + 121T^{2} \)
13 \( 1 + (3.67 - 3.67i)T - 169iT^{2} \)
17 \( 1 + (-11.3 - 11.3i)T + 289iT^{2} \)
19 \( 1 + 30.3iT - 361T^{2} \)
23 \( 1 + (-9.55 + 9.55i)T - 529iT^{2} \)
29 \( 1 - 15.3iT - 841T^{2} \)
31 \( 1 + 21.2T + 961T^{2} \)
37 \( 1 + (3.10 + 3.10i)T + 1.36e3iT^{2} \)
41 \( 1 + 18.2T + 1.68e3T^{2} \)
43 \( 1 + (19.4 - 19.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (-27.7 - 27.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (-56.8 + 56.8i)T - 2.80e3iT^{2} \)
59 \( 1 - 82iT - 3.48e3T^{2} \)
61 \( 1 - 94.5T + 3.72e3T^{2} \)
67 \( 1 + (-12.7 - 12.7i)T + 4.48e3iT^{2} \)
71 \( 1 - 77.7T + 5.04e3T^{2} \)
73 \( 1 + (90.2 - 90.2i)T - 5.32e3iT^{2} \)
79 \( 1 + 103. iT - 6.24e3T^{2} \)
83 \( 1 + (-22.8 + 22.8i)T - 6.88e3iT^{2} \)
89 \( 1 + 159. iT - 7.92e3T^{2} \)
97 \( 1 + (-56.7 - 56.7i)T + 9.40e3iT^{2} \)
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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

−8.942417184931799066374970763229, −8.499233617892990112208340023428, −7.21508196006075137495749772463, −6.85298777343701402634366466236, −5.85016144607590765239107067947, −4.92678824238729438810122872922, −4.04952563558638780087885239675, −3.13203376235943205349518852518, −1.93993741195009694329035882492, −0.74043426943753639741081455135, 0.905274722304856252344163841354, 2.03005754272346242522575066208, 3.33629825804808915535068965898, 3.99473963168527506903874379650, 5.17752803609039957936689399275, 5.86536083433017908167701274854, 6.79203127461152966822916512381, 7.55903974609895447419523013016, 8.320846676855008693214302751959, 9.237249190784667839788459560688

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