↑

Properties

Label 2-2160-9.2-c2-0-35
Degree $2$
Conductor $2160$
Sign $0.342 + 0.939i$
Analytic cond. $58.8557$
Root an. cond. $7.67174$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.93 − 1.11i)5-s + (1.87 − 3.24i)7-s + (−10.1 − 5.84i)11-s + (1.12 + 1.95i)13-s + 11.6i·17-s + 26.7·19-s + (−17.2 + 9.95i)23-s + (2.5 − 4.33i)25-s + (38.2 + 22.0i)29-s + (−26.1 − 45.2i)31-s − 8.37i·35-s + 14·37-s + (22.5 − 12.9i)41-s + (20.9 − 36.3i)43-s + (39.3 + 22.7i)47-s + ⋯
L(s)  = 1  + (0.387 − 0.223i)5-s + (0.267 − 0.463i)7-s + (−0.919 − 0.531i)11-s + (0.0866 + 0.150i)13-s + 0.687i·17-s + 1.40·19-s + (−0.749 + 0.432i)23-s + (0.100 − 0.173i)25-s + (1.31 + 0.761i)29-s + (−0.842 − 1.45i)31-s − 0.239i·35-s + 0.378·37-s + (0.548 − 0.316i)41-s + (0.488 − 0.845i)43-s + (0.837 + 0.483i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(58.8557\)
Root analytic conductor: \(7.67174\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1),\ 0.342 + 0.939i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.013754265\)
\(L(\frac12)\) \(\approx\) \(2.013754265\)
\(L(2)\) 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 \( 1 + (-1.93 + 1.11i)T \)
good7 \( 1 + (-1.87 + 3.24i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (10.1 + 5.84i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-1.12 - 1.95i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 11.6iT - 289T^{2} \)
19 \( 1 - 26.7T + 361T^{2} \)
23 \( 1 + (17.2 - 9.95i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-38.2 - 22.0i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (26.1 + 45.2i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 14T + 1.36e3T^{2} \)
41 \( 1 + (-22.5 + 12.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-20.9 + 36.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-39.3 - 22.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 10.8iT - 2.80e3T^{2} \)
59 \( 1 + (31.8 - 18.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-22.6 + 39.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (49.9 + 86.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 102. iT - 5.04e3T^{2} \)
73 \( 1 + 13.7T + 5.32e3T^{2} \)
79 \( 1 + (-28.3 + 49.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (78 + 45.0i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 95.2iT - 7.92e3T^{2} \)
97 \( 1 + (-50.8 + 88.0i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−8.797717051214860448729962659732, −7.76889661500876629379041815086, −7.49967824749738032436985576389, −6.19059065796740037700908590705, −5.64331975652631695699335635138, −4.77185458418120338753507080011, −3.81902219545204855343213895485, −2.83693312245337143217560867850, −1.70210117489154571244647799554, −0.56399570451897981737189089946, 1.05029815419755575348635292947, 2.37554572481287018688972237516, 2.95566761813608632407561024938, 4.28087313414391594029167260304, 5.21458186493688662759400567399, 5.70671187419513817706726543896, 6.78658686295519850274478152402, 7.50399674207900932491174321103, 8.234640624770566776679773852300, 9.073693993605804415023906060774

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