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Properties

Label 2-1620-9.4-c3-0-19
Degree $2$
Conductor $1620$
Sign $0.939 + 0.342i$
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 − 27.7i)7-s + (18 + 31.1i)11-s + (5 − 8.66i)13-s + 78·17-s + 140·19-s + (−96 + 166. i)23-s + (−12.5 − 21.6i)25-s + (3 + 5.19i)29-s + (8 − 13.8i)31-s − 160·35-s − 34·37-s + (−195 + 337. i)41-s + (26 + 45.0i)43-s + (204 + 353. i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.863 − 1.49i)7-s + (0.493 + 0.854i)11-s + (0.106 − 0.184i)13-s + 1.11·17-s + 1.69·19-s + (−0.870 + 1.50i)23-s + (−0.100 − 0.173i)25-s + (0.0192 + 0.0332i)29-s + (0.0463 − 0.0802i)31-s − 0.772·35-s − 0.151·37-s + (−0.742 + 1.28i)41-s + (0.0922 + 0.159i)43-s + (0.633 + 1.09i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.939 + 0.342i$
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: \((2,\ 1620,\ (\ :3/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.194391073\)
\(L(\frac12)\) \(\approx\) \(2.194391073\)
\(L(\frac{5}{2})\) 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 \( 1 + (16 + 27.7i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-18 - 31.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-5 + 8.66i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 78T + 4.91e3T^{2} \)
19 \( 1 - 140T + 6.85e3T^{2} \)
23 \( 1 + (96 - 166. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-8 + 13.8i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 34T + 5.06e4T^{2} \)
41 \( 1 + (195 - 337. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-26 - 45.0i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-204 - 353. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 114T + 1.48e5T^{2} \)
59 \( 1 + (-258 + 446. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-29 - 50.2i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-446 + 772. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 120T + 3.57e5T^{2} \)
73 \( 1 + 646T + 3.89e5T^{2} \)
79 \( 1 + (-584 - 1.01e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (366 + 633. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.59e3T + 7.04e5T^{2} \)
97 \( 1 + (97 + 168. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−9.355700014624795915165473210464, −7.87363911714823365878502758221, −7.50061521551121798022346313356, −6.66372111873311704582122268005, −5.73585567407966762357358433129, −4.81702235819604419058825297831, −3.78206363679394617051609795565, −3.23082141514231525811268273839, −1.54381115221371941539635474003, −0.78532214588856797542208634786, 0.69545023403847283485533183278, 2.16088284888887560299574468514, 3.04872974283581631192514055274, 3.73188987453581386795549364296, 5.30347347061979727225001769932, 5.80661186852532379381577915685, 6.49982454698800300044380698471, 7.41502357686374101558462391567, 8.559191282727349600772239756330, 8.957443219870005762100105087169

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