Properties

Label 2-1620-9.5-c2-0-27
Degree $2$
Conductor $1620$
Sign $-0.766 + 0.642i$
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 + (2 + 3.46i)7-s + (−5.80 + 3.35i)11-s + (3.5 − 6.06i)13-s − 20.1i·17-s + 8·19-s + (5.80 + 3.35i)23-s + (2.5 + 4.33i)25-s + (−40.6 + 23.4i)29-s + (−14.5 + 25.1i)31-s − 8.94i·35-s + 2·37-s + (11.6 + 6.70i)41-s + (3.5 + 6.06i)43-s + (−29.0 + 16.7i)47-s + ⋯
L(s)  = 1  + (−0.387 − 0.223i)5-s + (0.285 + 0.494i)7-s + (−0.528 + 0.304i)11-s + (0.269 − 0.466i)13-s − 1.18i·17-s + 0.421·19-s + (0.252 + 0.145i)23-s + (0.100 + 0.173i)25-s + (−1.40 + 0.809i)29-s + (−0.467 + 0.810i)31-s − 0.255i·35-s + 0.0540·37-s + (0.283 + 0.163i)41-s + (0.0813 + 0.140i)43-s + (−0.618 + 0.356i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(44.1418\)
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: \((2,\ 1620,\ (\ :1),\ -0.766 + 0.642i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6011942091\)
\(L(\frac12)\) \(\approx\) \(0.6011942091\)
\(L(2)\) 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 \( 1 + (-2 - 3.46i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (5.80 - 3.35i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-3.5 + 6.06i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 20.1iT - 289T^{2} \)
19 \( 1 - 8T + 361T^{2} \)
23 \( 1 + (-5.80 - 3.35i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (40.6 - 23.4i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (14.5 - 25.1i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 2T + 1.36e3T^{2} \)
41 \( 1 + (-11.6 - 6.70i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-3.5 - 6.06i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (29.0 - 16.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 93.9iT - 2.80e3T^{2} \)
59 \( 1 + (34.8 + 20.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (31 + 53.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-29 + 50.2i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 53.6iT - 5.04e3T^{2} \)
73 \( 1 + 52T + 5.32e3T^{2} \)
79 \( 1 + (-24.5 - 42.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-11.6 + 6.70i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 107. iT - 7.92e3T^{2} \)
97 \( 1 + (-17 - 29.4i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−8.980788358874368035089638317889, −8.008248774994903918047465002628, −7.46247314603207084790682853639, −6.55085647517169559684226120151, −5.22511620723386294947314992134, −5.10885599121157546219717218301, −3.70227473342916501889284490951, −2.83315268829907000053262748216, −1.60657944360340906418156716314, −0.16564842436604128501098413751, 1.29821439364242329130495703150, 2.53993188217001000921961169736, 3.75326914820335103540961405788, 4.30802335650193782142246720431, 5.53564619778701109705260881929, 6.22901091927213492414315077105, 7.34603335607952010867822225067, 7.78861011108665731417433374685, 8.675125383711637157389437904861, 9.481840026181639481223263903904

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