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Properties

Label 2-1620-9.4-c3-0-1
Degree $2$
Conductor $1620$
Sign $-0.173 - 0.984i$
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 + (−6.37 − 11.0i)7-s + (−17.3 − 30.0i)11-s + (37.8 − 65.5i)13-s − 84.2·17-s − 83.7·19-s + (−86.4 + 149. i)23-s + (−12.5 − 21.6i)25-s + (−44.1 − 76.4i)29-s + (−102. + 178. i)31-s − 63.7·35-s − 286·37-s + (160. − 278. i)41-s + (84.3 + 146. i)43-s + (195. + 338. i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.344 − 0.596i)7-s + (−0.476 − 0.824i)11-s + (0.807 − 1.39i)13-s − 1.20·17-s − 1.01·19-s + (−0.784 + 1.35i)23-s + (−0.100 − 0.173i)25-s + (−0.282 − 0.489i)29-s + (−0.595 + 1.03i)31-s − 0.307·35-s − 1.27·37-s + (0.612 − 1.06i)41-s + (0.299 + 0.518i)43-s + (0.605 + 1.04i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(95.5830\)
Root analytic conductor: \(9.77666\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :3/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2544749281\)
\(L(\frac12)\) \(\approx\) \(0.2544749281\)
\(L(\frac{5}{2})\) 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 \( 1 + (6.37 + 11.0i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (17.3 + 30.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-37.8 + 65.5i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 84.2T + 4.91e3T^{2} \)
19 \( 1 + 83.7T + 6.85e3T^{2} \)
23 \( 1 + (86.4 - 149. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (44.1 + 76.4i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (102. - 178. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 286T + 5.06e4T^{2} \)
41 \( 1 + (-160. + 278. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-84.3 - 146. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-195. - 338. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 91.6T + 1.48e5T^{2} \)
59 \( 1 + (61.3 - 106. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-439. - 760. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-518. + 898. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 605.T + 3.57e5T^{2} \)
73 \( 1 + 13.8T + 3.89e5T^{2} \)
79 \( 1 + (-286. - 496. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-313. - 543. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.01e3T + 7.04e5T^{2} \)
97 \( 1 + (577. + 1.00e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−9.116981327800988844729813408308, −8.483817756031752939462771627031, −7.77702821367294656318382745074, −6.81431767599381067770133061634, −5.88169999112761184941044873078, −5.34945763298443600282274309210, −4.07877596684079291874090883248, −3.40624145637500916718433915472, −2.20096459353017894294802824232, −0.909946398455254811384392327089, 0.06166694987873054348524209351, 2.00101792673370664386227220981, 2.34401690067858696178828121438, 3.85720525327020716913117335420, 4.50634399977694433780745867671, 5.65766047190934624814030154245, 6.58784140063515971461892662846, 6.87084033048464811441279964206, 8.164909074912607788986966956256, 8.885422505947626604409679506059

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