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Properties

Label 2-1800-5.2-c2-0-5
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  + (−7.67 − 7.67i)7-s − 7.79·11-s + (3.67 − 3.67i)13-s + (−3.34 − 3.34i)17-s + 28.3i·19-s + (14.4 − 14.4i)23-s − 43.3i·29-s − 40.7·31-s + (−12.8 − 12.8i)37-s − 37.7·41-s + (−36.5 + 36.5i)43-s + (52.2 + 52.2i)47-s + 68.7i·49-s + (47.1 − 47.1i)53-s + 82i·59-s + ⋯
L(s)  = 1  + (−1.09 − 1.09i)7-s − 0.708·11-s + (0.282 − 0.282i)13-s + (−0.196 − 0.196i)17-s + 1.49i·19-s + (0.628 − 0.628i)23-s − 1.49i·29-s − 1.31·31-s + (−0.348 − 0.348i)37-s − 0.921·41-s + (−0.850 + 0.850i)43-s + (1.11 + 1.11i)47-s + 1.40i·49-s + (0.888 − 0.888i)53-s + 1.38i·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

\(L(\frac{3}{2})\) \(\approx\) \(0.6980553924\)
\(L(\frac12)\) \(\approx\) \(0.6980553924\)
\(L(2)\) 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 \( 1 + (7.67 + 7.67i)T + 49iT^{2} \)
11 \( 1 + 7.79T + 121T^{2} \)
13 \( 1 + (-3.67 + 3.67i)T - 169iT^{2} \)
17 \( 1 + (3.34 + 3.34i)T + 289iT^{2} \)
19 \( 1 - 28.3iT - 361T^{2} \)
23 \( 1 + (-14.4 + 14.4i)T - 529iT^{2} \)
29 \( 1 + 43.3iT - 841T^{2} \)
31 \( 1 + 40.7T + 961T^{2} \)
37 \( 1 + (12.8 + 12.8i)T + 1.36e3iT^{2} \)
41 \( 1 + 37.7T + 1.68e3T^{2} \)
43 \( 1 + (36.5 - 36.5i)T - 1.84e3iT^{2} \)
47 \( 1 + (-52.2 - 52.2i)T + 2.20e3iT^{2} \)
53 \( 1 + (-47.1 + 47.1i)T - 2.80e3iT^{2} \)
59 \( 1 - 82iT - 3.48e3T^{2} \)
61 \( 1 - 55.4T + 3.72e3T^{2} \)
67 \( 1 + (-59.2 - 59.2i)T + 4.48e3iT^{2} \)
71 \( 1 - 58.2T + 5.04e3T^{2} \)
73 \( 1 + (21.7 - 21.7i)T - 5.32e3iT^{2} \)
79 \( 1 - 91.9iT - 6.24e3T^{2} \)
83 \( 1 + (-57.1 + 57.1i)T - 6.88e3iT^{2} \)
89 \( 1 + 120. iT - 7.92e3T^{2} \)
97 \( 1 + (-103. - 103. i)T + 9.40e3iT^{2} \)
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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

−9.411894472224361304375783934941, −8.399499108514272673689729033474, −7.64777934255433812713826957667, −6.92488178564329669447058695489, −6.12954373238104744816635625840, −5.29633881258149134366329058321, −4.08580922021335705783382740495, −3.50333729506415973106870876690, −2.40024426687312761044545813149, −0.905018552013800379320516530146, 0.21989879912165033152881117375, 1.94870855458313528813189251801, 2.92460517439856588655924208776, 3.65443841847620653974770641023, 5.13616608123869664978805160793, 5.48363138661119152388451439520, 6.72355911557401911320679707782, 7.02933822589386478979719168724, 8.345571107277109323913353980752, 9.024052607117377072242312660411

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