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Properties

Label 2-1620-9.7-c3-0-42
Degree $2$
Conductor $1620$
Sign $-0.642 + 0.766i$
Analytic cond. $95.5830$
Root an. cond. $9.77666$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.5 + 4.33i)5-s + (−8.28 + 14.3i)7-s + (36.2 − 62.8i)11-s + (29.9 + 51.8i)13-s + 15.8·17-s − 136.·19-s + (−81.5 − 141. i)23-s + (−12.5 + 21.6i)25-s + (−5.79 + 10.0i)29-s + (20.5 + 35.5i)31-s − 82.8·35-s − 242.·37-s + (−28.9 − 50.1i)41-s + (132. − 229. i)43-s + (−299. + 519. i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (−0.447 + 0.774i)7-s + (0.994 − 1.72i)11-s + (0.638 + 1.10i)13-s + 0.226·17-s − 1.64·19-s + (−0.739 − 1.28i)23-s + (−0.100 + 0.173i)25-s + (−0.0371 + 0.0643i)29-s + (0.118 + 0.205i)31-s − 0.400·35-s − 1.07·37-s + (−0.110 − 0.190i)41-s + (0.469 − 0.812i)43-s + (−0.930 + 1.61i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(95.5830\)
Root analytic conductor: \(9.77666\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :3/2),\ -0.642 + 0.766i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5721939670\)
\(L(\frac12)\) \(\approx\) \(0.5721939670\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.5 - 4.33i)T \)
good7 \( 1 + (8.28 - 14.3i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-36.2 + 62.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-29.9 - 51.8i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 15.8T + 4.91e3T^{2} \)
19 \( 1 + 136.T + 6.85e3T^{2} \)
23 \( 1 + (81.5 + 141. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (5.79 - 10.0i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-20.5 - 35.5i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 242.T + 5.06e4T^{2} \)
41 \( 1 + (28.9 + 50.1i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-132. + 229. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (299. - 519. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 592.T + 1.48e5T^{2} \)
59 \( 1 + (144. + 250. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (412. - 714. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (405. + 701. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 966.T + 3.57e5T^{2} \)
73 \( 1 - 802.T + 3.89e5T^{2} \)
79 \( 1 + (-581. + 1.00e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (251. - 435. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 8.76T + 7.04e5T^{2} \)
97 \( 1 + (69.1 - 119. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−8.852397628365421997803701078561, −8.239829121946256470541546950029, −6.82119644082157477839723885690, −6.21476568363547360888523568557, −5.88215409711193176338558536828, −4.39787061611163587081184725024, −3.61108712207107702319951541246, −2.63399967771650999088559359768, −1.56735910509166126831002550183, −0.12315630611638226828627507252, 1.21347399405289743638546424089, 2.11488495609284122998472432847, 3.62382048970794020918452682388, 4.16757971346931252147786700765, 5.18185405342443567994619366597, 6.19108224751819781906805115474, 6.89570686868831348465970435142, 7.68199667409678572705725254616, 8.546830756440268083991175569291, 9.413931744592779956292595929402

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