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Properties

Label 2-1620-9.4-c3-0-24
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 + (−1 − 1.73i)7-s + (15 + 25.9i)11-s + (2 − 3.46i)13-s − 90·17-s − 28·19-s + (60 − 103. i)23-s + (−12.5 − 21.6i)25-s + (105 + 181. i)29-s + (2 − 3.46i)31-s − 10·35-s + 200·37-s + (120 − 207. i)41-s + (68 + 117. i)43-s + (−60 − 103. i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.0539 − 0.0935i)7-s + (0.411 + 0.712i)11-s + (0.0426 − 0.0739i)13-s − 1.28·17-s − 0.338·19-s + (0.543 − 0.942i)23-s + (−0.100 − 0.173i)25-s + (0.672 + 1.16i)29-s + (0.0115 − 0.0200i)31-s − 0.0482·35-s + 0.888·37-s + (0.457 − 0.791i)41-s + (0.241 + 0.417i)43-s + (−0.186 − 0.322i)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

\(L(2)\) \(\approx\) \(2.097660147\)
\(L(\frac12)\) \(\approx\) \(2.097660147\)
\(L(\frac{5}{2})\) 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 \( 1 + (1 + 1.73i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-15 - 25.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 90T + 4.91e3T^{2} \)
19 \( 1 + 28T + 6.85e3T^{2} \)
23 \( 1 + (-60 + 103. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-105 - 181. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 200T + 5.06e4T^{2} \)
41 \( 1 + (-120 + 207. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-68 - 117. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (60 + 103. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 30T + 1.48e5T^{2} \)
59 \( 1 + (225 - 389. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-83 - 143. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (454 - 786. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 1.02e3T + 3.57e5T^{2} \)
73 \( 1 + 250T + 3.89e5T^{2} \)
79 \( 1 + (-458 - 793. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (570 + 987. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 420T + 7.04e5T^{2} \)
97 \( 1 + (769 + 1.33e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−8.883759943364357859218304927373, −8.446319392553720728857708284538, −7.20050436109476826106362353364, −6.69634879381948113470005802642, −5.73808956093631785789058529017, −4.68188998467233301478501893869, −4.17237863640150590573124367733, −2.80725673286772482099290753093, −1.85016471102593187683118742496, −0.65143256734211370247081735996, 0.74324754200861791421041994190, 2.05587660198820689938867939007, 2.99694672027115655748303861153, 4.01123908595908201391263307822, 4.90673438356624743480239432816, 6.10246965044580990215061873354, 6.43978269491275225685642895930, 7.49537581278123381593460022759, 8.303412213753716073593860681159, 9.179481513238180583526154089039

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