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Properties

Label 2-1620-9.7-c3-0-34
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 + (1.55 − 2.68i)7-s + (20.2 − 35.0i)11-s + (16.7 + 28.9i)13-s + 70.5·17-s + 146.·19-s + (−50.2 − 87.0i)23-s + (−12.5 + 21.6i)25-s + (−50.2 + 87.0i)29-s + (12.9 + 22.4i)31-s − 15.5·35-s − 218.·37-s + (46.5 + 80.6i)41-s + (−8.63 + 14.9i)43-s + (−206. + 357. i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (0.0838 − 0.145i)7-s + (0.555 − 0.961i)11-s + (0.356 + 0.617i)13-s + 1.00·17-s + 1.76·19-s + (−0.455 − 0.789i)23-s + (−0.100 + 0.173i)25-s + (−0.321 + 0.557i)29-s + (0.0750 + 0.129i)31-s − 0.0749·35-s − 0.971·37-s + (0.177 + 0.307i)41-s + (−0.0306 + 0.0530i)43-s + (−0.640 + 1.10i)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

\(L(2)\) \(\approx\) \(2.388749543\)
\(L(\frac12)\) \(\approx\) \(2.388749543\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.55 + 2.68i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-20.2 + 35.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-16.7 - 28.9i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 70.5T + 4.91e3T^{2} \)
19 \( 1 - 146.T + 6.85e3T^{2} \)
23 \( 1 + (50.2 + 87.0i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (50.2 - 87.0i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-12.9 - 22.4i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 218.T + 5.06e4T^{2} \)
41 \( 1 + (-46.5 - 80.6i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (8.63 - 14.9i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (206. - 357. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 471.T + 1.48e5T^{2} \)
59 \( 1 + (321. + 556. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (245. - 424. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (218. + 378. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.11e3T + 3.57e5T^{2} \)
73 \( 1 - 222.T + 3.89e5T^{2} \)
79 \( 1 + (-201. + 349. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-330. + 571. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 173.T + 7.04e5T^{2} \)
97 \( 1 + (-295. + 510. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−8.998647008719550503136079351829, −8.137602189783427080495199235753, −7.46291977429088075249079750515, −6.47715581278334605242972451699, −5.67104664148513234714497972993, −4.83099373519599163331704781349, −3.76083967485491057766468119154, −3.09957054149778554872326080061, −1.52973220953015020772436594560, −0.71081366311811625446271859026, 0.881368529721790449658195195640, 1.99559088519357962679497357648, 3.26377679197488960829707892714, 3.87375774338204350875735033111, 5.16668772822770495107353162723, 5.73007749789249656324601020149, 6.88890117694245774738726633997, 7.50277791069195186123831595254, 8.171447191977155380544876871155, 9.269590656471205883362266035925

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