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Properties

Label 2-1620-45.34-c1-0-15
Degree $2$
Conductor $1620$
Sign $0.883 - 0.468i$
Analytic cond. $12.9357$
Root an. cond. $3.59663$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.308 + 2.21i)5-s + (4.27 + 2.46i)7-s + (1.20 − 2.08i)11-s + (2.51 − 1.45i)13-s − 6.86i·17-s + 4.17·19-s + (2.90 − 1.67i)23-s + (−4.80 + 1.36i)25-s + (2.59 − 4.5i)29-s + (3.08 + 5.35i)31-s + (−4.14 + 10.2i)35-s − 7.84i·37-s + (2.93 + 5.08i)41-s + (−4.27 − 2.46i)43-s + (−10.3 − 5.95i)47-s + ⋯
L(s)  = 1  + (0.138 + 0.990i)5-s + (1.61 + 0.932i)7-s + (0.363 − 0.629i)11-s + (0.698 − 0.403i)13-s − 1.66i·17-s + 0.958·19-s + (0.606 − 0.350i)23-s + (−0.961 + 0.273i)25-s + (0.482 − 0.835i)29-s + (0.554 + 0.961i)31-s + (−0.700 + 1.72i)35-s − 1.28i·37-s + (0.458 + 0.794i)41-s + (−0.651 − 0.376i)43-s + (−1.50 − 0.868i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.883 - 0.468i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ 0.883 - 0.468i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.341389876\)
\(L(\frac12)\) \(\approx\) \(2.341389876\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.308 - 2.21i)T \)
good7 \( 1 + (-4.27 - 2.46i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.20 + 2.08i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.51 + 1.45i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.86iT - 17T^{2} \)
19 \( 1 - 4.17T + 19T^{2} \)
23 \( 1 + (-2.90 + 1.67i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.59 + 4.5i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.08 - 5.35i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.84iT - 37T^{2} \)
41 \( 1 + (-2.93 - 5.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.27 + 2.46i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (10.3 + 5.95i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.54iT - 53T^{2} \)
59 \( 1 + (0.525 + 0.910i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.58 - 7.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.50 - 2.02i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 - 2.02iT - 73T^{2} \)
79 \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.49 - 2.59i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.09T + 89T^{2} \)
97 \( 1 + (-0.764 - 0.441i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.321557578182991423631258327871, −8.659130019093059080724265263629, −7.86442840656696216529411914228, −7.15379774285037595464904300807, −6.12429020590088879866274821370, −5.39732928430481267185242724750, −4.61019315754022626492346038303, −3.22166577538490285742243658019, −2.52896332803372647864831788050, −1.22042631456308810225506503996, 1.30074748956038411834469220629, 1.64446217330760688797560269864, 3.60208271303249083282870283839, 4.49810809932909906376065160102, 4.93757089679416917011854679376, 6.02493337196784446309241527351, 7.02567416904860823506675966486, 8.047197575669937802455277186787, 8.292946592572258349932101292231, 9.278881822588788919218843421922

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