Properties

Label 2-1620-9.5-c2-0-29
Degree $2$
Conductor $1620$
Sign $-0.0871 + 0.996i$
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  + (1.93 + 1.11i)5-s + (−0.973 − 1.68i)7-s + (13.6 − 7.90i)11-s + (11.4 − 19.8i)13-s − 21.9i·17-s + 16.8·19-s + (−24.2 − 13.9i)23-s + (2.5 + 4.33i)25-s + (−36.0 + 20.8i)29-s + (−25.9 + 45.0i)31-s − 4.35i·35-s − 49.8·37-s + (52.0 + 30.0i)41-s + (−25.1 − 43.5i)43-s + (−61.6 + 35.5i)47-s + ⋯
L(s)  = 1  + (0.387 + 0.223i)5-s + (−0.139 − 0.240i)7-s + (1.24 − 0.719i)11-s + (0.882 − 1.52i)13-s − 1.29i·17-s + 0.885·19-s + (−1.05 − 0.608i)23-s + (0.100 + 0.173i)25-s + (−1.24 + 0.717i)29-s + (−0.838 + 1.45i)31-s − 0.124i·35-s − 1.34·37-s + (1.26 + 0.733i)41-s + (−0.584 − 1.01i)43-s + (−1.31 + 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.0871 + 0.996i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ -0.0871 + 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.965634464\)
\(L(\frac12)\) \(\approx\) \(1.965634464\)
\(L(2)\) 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 \( 1 + (-1.93 - 1.11i)T \)
good7 \( 1 + (0.973 + 1.68i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-13.6 + 7.90i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-11.4 + 19.8i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 21.9iT - 289T^{2} \)
19 \( 1 - 16.8T + 361T^{2} \)
23 \( 1 + (24.2 + 13.9i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (36.0 - 20.8i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (25.9 - 45.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 49.8T + 1.36e3T^{2} \)
41 \( 1 + (-52.0 - 30.0i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (25.1 + 43.5i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (61.6 - 35.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 21.7iT - 2.80e3T^{2} \)
59 \( 1 + (-34.4 - 19.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (53.7 + 93.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-40.9 + 70.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 62.1iT - 5.04e3T^{2} \)
73 \( 1 - 46.9T + 5.32e3T^{2} \)
79 \( 1 + (0.00315 + 0.00546i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-55.5 + 32.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 51.0iT - 7.92e3T^{2} \)
97 \( 1 + (22.3 + 38.6i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.066100995783203324260255906282, −8.265461743951426646724754774739, −7.33890445978592115425787699023, −6.56067017677611566617211711714, −5.74392652079976594150971673520, −5.03392161248960812711342433337, −3.54698022473935853215365145649, −3.22871803219495785864735802497, −1.63905008290065987051585179821, −0.53270515890645494349047252049, 1.47849682618318178401307776284, 2.02930754862722744382587502287, 3.91745588784516630872019239040, 4.00023446855887737454835874610, 5.51479554075749160660050975170, 6.17024328003958000387832398995, 6.89479063312642348010134683920, 7.84657152271192684795706271685, 8.831537330928276081129417962206, 9.426483131284219460510343165050

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