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Properties

Label 2-1620-9.4-c3-0-8
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 + (−16.8 − 29.1i)7-s + (−4.74 − 8.22i)11-s + (−19.8 + 34.3i)13-s − 106.·17-s − 96.5·19-s + (20.0 − 34.6i)23-s + (−12.5 − 21.6i)25-s + (126. + 218. i)29-s + (−24.4 + 42.3i)31-s − 168.·35-s − 136.·37-s + (69.5 − 120. i)41-s + (101. + 175. i)43-s + (150. + 259. i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.909 − 1.57i)7-s + (−0.130 − 0.225i)11-s + (−0.423 + 0.733i)13-s − 1.51·17-s − 1.16·19-s + (0.181 − 0.314i)23-s + (−0.100 − 0.173i)25-s + (0.808 + 1.39i)29-s + (−0.141 + 0.245i)31-s − 0.813·35-s − 0.606·37-s + (0.264 − 0.458i)41-s + (0.358 + 0.621i)43-s + (0.465 + 0.806i)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: $\chi_{1620} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :3/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8207275388\)
\(L(\frac12)\) \(\approx\) \(0.8207275388\)
\(L(\frac{5}{2})\) 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 \( 1 + (16.8 + 29.1i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (4.74 + 8.22i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (19.8 - 34.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 106.T + 4.91e3T^{2} \)
19 \( 1 + 96.5T + 6.85e3T^{2} \)
23 \( 1 + (-20.0 + 34.6i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-126. - 218. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (24.4 - 42.3i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 136.T + 5.06e4T^{2} \)
41 \( 1 + (-69.5 + 120. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-101. - 175. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-150. - 259. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 344.T + 1.48e5T^{2} \)
59 \( 1 + (30.4 - 52.7i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-61.0 - 105. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-372. + 644. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 436.T + 3.57e5T^{2} \)
73 \( 1 - 586.T + 3.89e5T^{2} \)
79 \( 1 + (236. + 410. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (499. + 865. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 204.T + 7.04e5T^{2} \)
97 \( 1 + (-816. - 1.41e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−9.071136415352080737339081034030, −8.512297161857726672028463589528, −7.24716605505514385232161816488, −6.81855819700399602751702554629, −6.09104227155947717807156428416, −4.66804922008152497248308555005, −4.26692996345239900189696626756, −3.18058234027122998657035786221, −1.97487690371201575242372545517, −0.70200045830276321285471631959, 0.24518963504275157800621430020, 2.35284044220657609937701660252, 2.45422041817290260652168076528, 3.77837751325313850283036367636, 4.92133486709111223845216680978, 5.84448276001467470936685290865, 6.39194346716146437874223468898, 7.19738929703902738247619326464, 8.418268792987705681976672942805, 8.859288581579308516073015637591

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