Properties

Label 2-1620-9.7-c3-0-22
Degree $2$
Conductor $1620$
Sign $0.766 + 0.642i$
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 + (−8.05 + 13.9i)7-s + (−27.7 + 48.0i)11-s + (−21.7 − 37.6i)13-s − 25.5·17-s − 103.·19-s + (−2.23 − 3.87i)23-s + (−12.5 + 21.6i)25-s + (−2.23 + 3.87i)29-s + (22.5 + 39.0i)31-s + 80.5·35-s + 69.5·37-s + (−241. − 418. i)41-s + (−75.8 + 131. i)43-s + (33.8 − 58.5i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.434 + 0.753i)7-s + (−0.760 + 1.31i)11-s + (−0.463 − 0.802i)13-s − 0.364·17-s − 1.24·19-s + (−0.0202 − 0.0351i)23-s + (−0.100 + 0.173i)25-s + (−0.0143 + 0.0248i)29-s + (0.130 + 0.226i)31-s + 0.388·35-s + 0.309·37-s + (−0.920 − 1.59i)41-s + (−0.269 + 0.466i)43-s + (0.104 − 0.181i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.766 + 0.642i$
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: \((2,\ 1620,\ (\ :3/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9155804045\)
\(L(\frac12)\) \(\approx\) \(0.9155804045\)
\(L(\frac{5}{2})\) 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 \( 1 + (8.05 - 13.9i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (27.7 - 48.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (21.7 + 37.6i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 25.5T + 4.91e3T^{2} \)
19 \( 1 + 103.T + 6.85e3T^{2} \)
23 \( 1 + (2.23 + 3.87i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (2.23 - 3.87i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-22.5 - 39.0i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 69.5T + 5.06e4T^{2} \)
41 \( 1 + (241. + 418. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (75.8 - 131. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-33.8 + 58.5i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 278.T + 1.48e5T^{2} \)
59 \( 1 + (128. + 223. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-244. + 423. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-386. - 669. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 536.T + 3.57e5T^{2} \)
73 \( 1 + 65.5T + 3.89e5T^{2} \)
79 \( 1 + (374. - 648. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (630. - 1.09e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.32e3T + 7.04e5T^{2} \)
97 \( 1 + (-16.4 + 28.5i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−8.855386239277317445441402964501, −8.240952660113773110957261096171, −7.36077304167329075473356468876, −6.58613469227921919378224663866, −5.52328543625860939335970644463, −4.89899233156002580901529838420, −3.95352472336723462470569207640, −2.68487171441241336226363608888, −1.99170877021759190329343580410, −0.31620939483641639636445961774, 0.60644622250810316576767405203, 2.15187702896376035357717661801, 3.15733138707328541727610129701, 4.00954302176275887816087713611, 4.89618407160046825574719332960, 6.09110405117059757715037826988, 6.65275524284203723242492456691, 7.51238032763010826276184541778, 8.316415998722797477994501555819, 9.033306970602882471049710130075

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