Properties

Label 2-1620-9.7-c3-0-33
Degree $2$
Conductor $1620$
Sign $-0.173 + 0.984i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.5 + 4.33i)5-s + (−14.9 + 25.9i)7-s + (−7.95 + 13.7i)11-s + (−20.9 − 36.3i)13-s − 36.9·17-s − 10.9·19-s + (57.3 + 99.3i)23-s + (−12.5 + 21.6i)25-s + (−72.7 + 126. i)29-s + (23.2 + 40.2i)31-s − 149.·35-s + 74·37-s + (−167. − 290. i)41-s + (−42.2 + 73.0i)43-s + (−127. + 221. i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (−0.807 + 1.39i)7-s + (−0.218 + 0.377i)11-s + (−0.447 − 0.774i)13-s − 0.526·17-s − 0.131·19-s + (0.519 + 0.900i)23-s + (−0.100 + 0.173i)25-s + (−0.466 + 0.807i)29-s + (0.134 + 0.233i)31-s − 0.722·35-s + 0.328·37-s + (−0.638 − 1.10i)41-s + (−0.149 + 0.259i)43-s + (−0.396 + 0.685i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.173 + 0.984i$
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: \((2,\ 1620,\ (\ :3/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1552272908\)
\(L(\frac12)\) \(\approx\) \(0.1552272908\)
\(L(\frac{5}{2})\) 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 \( 1 + (14.9 - 25.9i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (7.95 - 13.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (20.9 + 36.3i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 36.9T + 4.91e3T^{2} \)
19 \( 1 + 10.9T + 6.85e3T^{2} \)
23 \( 1 + (-57.3 - 99.3i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (72.7 - 126. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-23.2 - 40.2i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 74T + 5.06e4T^{2} \)
41 \( 1 + (167. + 290. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (42.2 - 73.0i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (127. - 221. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 399.T + 1.48e5T^{2} \)
59 \( 1 + (1.28 + 2.21i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (284. - 492. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (384. + 666. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 441.T + 3.57e5T^{2} \)
73 \( 1 - 966.T + 3.89e5T^{2} \)
79 \( 1 + (251. - 435. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-635. + 1.10e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 0.240T + 7.04e5T^{2} \)
97 \( 1 + (-3.35 + 5.81i)T + (-4.56e5 - 7.90e5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.036103254483313411055913817613, −8.024334859493719443080102697705, −7.14344878684584039546824561887, −6.32834440727651234907312010032, −5.57346628868798261164943666659, −4.87469662850903091398053194819, −3.40096487614082943402396712019, −2.75902254570154864297032359171, −1.81005599507184486496156518408, −0.03927333388567705552260916062, 0.877849678506526531314915767479, 2.19817414366813298802530399118, 3.37725204880363657667892370204, 4.26865044784176658031509555054, 4.96000249205269303861416986305, 6.28308730280209992247996623124, 6.72463597507564743970126935014, 7.62175543168577851093842265426, 8.455834109224853997788446036953, 9.424966519881626039115676744839

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