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Properties

Label 2-1620-45.29-c2-0-4
Degree $2$
Conductor $1620$
Sign $-0.772 + 0.634i$
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  + (−2.15 + 4.50i)5-s + (6.51 + 3.76i)7-s + (5.77 + 3.33i)11-s + (−13.4 + 7.78i)13-s − 18.9·17-s − 27.1·19-s + (9.80 + 16.9i)23-s + (−15.6 − 19.4i)25-s + (−6.03 − 3.48i)29-s + (−14.7 − 25.5i)31-s + (−31.0 + 21.2i)35-s − 52.2i·37-s + (52.7 − 30.4i)41-s + (9.23 + 5.33i)43-s + (−37.9 + 65.6i)47-s + ⋯
L(s)  = 1  + (−0.431 + 0.901i)5-s + (0.931 + 0.537i)7-s + (0.525 + 0.303i)11-s + (−1.03 + 0.598i)13-s − 1.11·17-s − 1.42·19-s + (0.426 + 0.738i)23-s + (−0.627 − 0.778i)25-s + (−0.208 − 0.120i)29-s + (−0.476 − 0.824i)31-s + (−0.887 + 0.607i)35-s − 1.41i·37-s + (1.28 − 0.742i)41-s + (0.214 + 0.123i)43-s + (−0.806 + 1.39i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 + 0.634i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.772 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.772 + 0.634i$
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: \((2,\ 1620,\ (\ :1),\ -0.772 + 0.634i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2283996813\)
\(L(\frac12)\) \(\approx\) \(0.2283996813\)
\(L(2)\) 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 \( 1 + (2.15 - 4.50i)T \)
good7 \( 1 + (-6.51 - 3.76i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-5.77 - 3.33i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (13.4 - 7.78i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 18.9T + 289T^{2} \)
19 \( 1 + 27.1T + 361T^{2} \)
23 \( 1 + (-9.80 - 16.9i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (6.03 + 3.48i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (14.7 + 25.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 52.2iT - 1.36e3T^{2} \)
41 \( 1 + (-52.7 + 30.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-9.23 - 5.33i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (37.9 - 65.6i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 47.3T + 2.80e3T^{2} \)
59 \( 1 + (21.1 - 12.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (20.3 - 35.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-51.0 + 29.4i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 33.9iT - 5.04e3T^{2} \)
73 \( 1 + 79.8iT - 5.32e3T^{2} \)
79 \( 1 + (43.0 - 74.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-76.5 + 132. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 83.3iT - 7.92e3T^{2} \)
97 \( 1 + (-14.0 - 8.10i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.496880601299786533602349601520, −9.007705505638925893968684154767, −7.984069778892100661184773448596, −7.30510809453556885033575202157, −6.60450547280541861411223374606, −5.68131003083235561419969044232, −4.51485601731652938198931819148, −4.00948521777027099235457459114, −2.50629850601157494193524154753, −1.94803313009552692557411103355, 0.06130952129473960717161134787, 1.23202113577567758794724080530, 2.39674564623579908618329296648, 3.83659830638641845384638118582, 4.63526906893270246097426898129, 5.09197779043761673263315226162, 6.38443403382544462559155881216, 7.19124190101830256505993877921, 8.135973909082611460417623132505, 8.558585898361144672158273090976

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