Properties

Label 2-1620-9.2-c2-0-19
Degree $2$
Conductor $1620$
Sign $-0.173 + 0.984i$
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 + (−5.85 + 10.1i)7-s + (−8.40 − 4.85i)11-s + (0.145 + 0.252i)13-s + 15i·17-s + 12.4·19-s + (−29.8 + 17.2i)23-s + (2.5 − 4.33i)25-s + (−3.21 − 1.85i)29-s + (−3.20 − 5.55i)31-s − 26.1i·35-s + 24.2·37-s + (21.2 − 12.2i)41-s + (−37.9 + 65.7i)43-s + (−22.0 − 12.7i)47-s + ⋯
L(s)  = 1  + (−0.387 + 0.223i)5-s + (−0.836 + 1.44i)7-s + (−0.764 − 0.441i)11-s + (0.0112 + 0.0194i)13-s + 0.882i·17-s + 0.653·19-s + (−1.29 + 0.748i)23-s + (0.100 − 0.173i)25-s + (−0.110 − 0.0639i)29-s + (−0.103 − 0.179i)31-s − 0.748i·35-s + 0.655·37-s + (0.518 − 0.299i)41-s + (−0.883 + 1.52i)43-s + (−0.468 − 0.270i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ -0.173 + 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2237447481\)
\(L(\frac12)\) \(\approx\) \(0.2237447481\)
\(L(2)\) 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 \( 1 + (5.85 - 10.1i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (8.40 + 4.85i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-0.145 - 0.252i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 15iT - 289T^{2} \)
19 \( 1 - 12.4T + 361T^{2} \)
23 \( 1 + (29.8 - 17.2i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (3.21 + 1.85i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (3.20 + 5.55i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 24.2T + 1.36e3T^{2} \)
41 \( 1 + (-21.2 + 12.2i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (37.9 - 65.7i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (22.0 + 12.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 64.0iT - 2.80e3T^{2} \)
59 \( 1 + (-61.5 + 35.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (39.7 - 68.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (7 + 12.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 91.9iT - 5.04e3T^{2} \)
73 \( 1 - 13.1T + 5.32e3T^{2} \)
79 \( 1 + (-73.7 + 127. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (121. + 70.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 105. iT - 7.92e3T^{2} \)
97 \( 1 + (7.70 - 13.3i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−8.972993380355989719504541411797, −8.183899653369577409263470783933, −7.56620442188217339745842869614, −6.27330382494607472946002660858, −5.91524804169150639272808168832, −4.97720397705409124398093075953, −3.66219854254286032221950680048, −2.95879133530021215186369600474, −1.94178855512327625494667991152, −0.07320100026357682183324408179, 0.913098598141007074683826303175, 2.52002090131086952960280345190, 3.58608286637955001075952130438, 4.32522966079391884305259710147, 5.21786376689820510535153707848, 6.33902072834000751569968634224, 7.21625137765599650827454213195, 7.62834524678183262262842754025, 8.560988070635878001166900666119, 9.712342399821704339930091806383

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