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Properties

Label 2-1620-9.4-c3-0-30
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 + (−3.10 − 5.38i)7-s + (−14.1 − 24.5i)11-s + (4.30 − 7.45i)13-s + 90.1·17-s + 114.·19-s + (−24.2 + 41.9i)23-s + (−12.5 − 21.6i)25-s + (152. + 264. i)29-s + (−46.7 + 81.0i)31-s − 31.0·35-s − 282.·37-s + (−14.3 + 24.9i)41-s + (177. + 306. i)43-s + (−261. − 452. i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.167 − 0.290i)7-s + (−0.388 − 0.672i)11-s + (0.0917 − 0.158i)13-s + 1.28·17-s + 1.37·19-s + (−0.219 + 0.380i)23-s + (−0.100 − 0.173i)25-s + (0.979 + 1.69i)29-s + (−0.271 + 0.469i)31-s − 0.150·35-s − 1.25·37-s + (−0.0547 + 0.0949i)41-s + (0.628 + 1.08i)43-s + (−0.811 − 1.40i)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: $\chi_{1620} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :3/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.288560304\)
\(L(\frac12)\) \(\approx\) \(2.288560304\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.10 + 5.38i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (14.1 + 24.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-4.30 + 7.45i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 90.1T + 4.91e3T^{2} \)
19 \( 1 - 114.T + 6.85e3T^{2} \)
23 \( 1 + (24.2 - 41.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-152. - 264. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (46.7 - 81.0i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 282.T + 5.06e4T^{2} \)
41 \( 1 + (14.3 - 24.9i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-177. - 306. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (261. + 452. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 66.9T + 1.48e5T^{2} \)
59 \( 1 + (3.81 - 6.60i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (4.70 + 8.15i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-247. + 428. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 560.T + 3.57e5T^{2} \)
73 \( 1 - 1.11e3T + 3.89e5T^{2} \)
79 \( 1 + (520. + 902. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-22.7 - 39.4i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 357.T + 7.04e5T^{2} \)
97 \( 1 + (-60.0 - 103. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−8.884969587289266907964599722126, −8.151082487958222809134906675076, −7.38485779716722548105331328179, −6.51879691856047277772697827100, −5.37619269680745957984221175655, −5.13263838808933523115681258071, −3.59177342949906912257989344451, −3.08800214933276363293917393563, −1.54606654370334288304071191434, −0.65206543728855847308751472942, 0.883425236583827624514357334208, 2.17621111643563351853602878442, 3.04051552668154476355734188885, 4.04945777711299159496176444174, 5.17523144703093010992348239830, 5.82658142015831905093236325916, 6.76256620860905157310506206999, 7.59254213230313889035521600919, 8.193270062757331084641221569432, 9.383132824449416119500270618563

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