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Properties

Label 2-1620-45.29-c2-0-27
Degree $2$
Conductor $1620$
Sign $0.687 - 0.726i$
Analytic cond. $44.1418$
Root an. cond. $6.64392$
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  + (−2.75 + 4.17i)5-s + (6.92 + 4i)7-s + (7.74 + 4.47i)11-s + (10.3 − 6i)13-s + 31.3·17-s − 6·19-s + (2.23 + 3.87i)23-s + (−9.82 − 22.9i)25-s + (23.2 + 13.4i)29-s + (−17 − 29.4i)31-s + (−35.7 + 17.8i)35-s − 44i·37-s + (15.4 − 8.94i)41-s + (24.2 + 14i)43-s + (2.23 − 3.87i)47-s + ⋯
L(s)  = 1  + (−0.550 + 0.834i)5-s + (0.989 + 0.571i)7-s + (0.704 + 0.406i)11-s + (0.799 − 0.461i)13-s + 1.84·17-s − 0.315·19-s + (0.0972 + 0.168i)23-s + (−0.392 − 0.919i)25-s + (0.801 + 0.462i)29-s + (−0.548 − 0.949i)31-s + (−1.02 + 0.511i)35-s − 1.18i·37-s + (0.377 − 0.218i)41-s + (0.563 + 0.325i)43-s + (0.0475 − 0.0824i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.687 - 0.726i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ 0.687 - 0.726i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.449196519\)
\(L(\frac12)\) \(\approx\) \(2.449196519\)
\(L(2)\) 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 \( 1 + (2.75 - 4.17i)T \)
good7 \( 1 + (-6.92 - 4i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-7.74 - 4.47i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-10.3 + 6i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 31.3T + 289T^{2} \)
19 \( 1 + 6T + 361T^{2} \)
23 \( 1 + (-2.23 - 3.87i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-23.2 - 13.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (17 + 29.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 44iT - 1.36e3T^{2} \)
41 \( 1 + (-15.4 + 8.94i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-24.2 - 14i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-2.23 + 3.87i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 40.2T + 2.80e3T^{2} \)
59 \( 1 + (-85.2 + 49.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (37 - 64.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-79.6 + 46i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 53.6iT - 5.04e3T^{2} \)
73 \( 1 - 56iT - 5.32e3T^{2} \)
79 \( 1 + (-39 + 67.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (51.4 - 89.0i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 17.8iT - 7.92e3T^{2} \)
97 \( 1 + (27.7 + 16i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.295506468957732026641039787252, −8.311341649133653270911351088279, −7.81921624288101457140036098600, −7.01530708821963748090638873242, −6.01196376707672780863199384278, −5.31275964005543146128764724149, −4.12882239101776982528396021269, −3.39712503917245192754917286265, −2.25823820793768436679213305407, −1.04182500290628080580830634959, 0.914517795649753385210500789644, 1.50335977623397322346618022703, 3.30993111547073370092424224781, 4.10183801183203579700445325651, 4.85685665047214013704912666548, 5.74295139530026612328413801049, 6.75296373746511378704844075417, 7.75846081853016757829082255044, 8.260382239816054738480776919785, 8.924472349574692766711806331306

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