Properties

Label 2-2160-1.1-c3-0-0
Degree $2$
Conductor $2160$
Sign $1$
Analytic cond. $127.444$
Root an. cond. $11.2891$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·5-s − 23.9·7-s − 57.9·11-s − 8.16·13-s − 50.0·17-s − 69.7·19-s − 4.92·23-s + 25·25-s + 79.4·29-s − 260.·31-s + 119.·35-s − 223.·37-s − 337.·41-s − 326.·43-s − 89.6·47-s + 229.·49-s − 543.·53-s + 289.·55-s − 92·59-s + 159.·61-s + 40.8·65-s + 910.·67-s + 293.·71-s + 142.·73-s + 1.38e3·77-s − 1.10e3·79-s − 813.·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.29·7-s − 1.58·11-s − 0.174·13-s − 0.714·17-s − 0.842·19-s − 0.0446·23-s + 0.200·25-s + 0.508·29-s − 1.50·31-s + 0.577·35-s − 0.994·37-s − 1.28·41-s − 1.15·43-s − 0.278·47-s + 0.668·49-s − 1.40·53-s + 0.710·55-s − 0.203·59-s + 0.334·61-s + 0.0778·65-s + 1.66·67-s + 0.490·71-s + 0.227·73-s + 2.05·77-s − 1.57·79-s − 1.07·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $1$
Analytic conductor: \(127.444\)
Root analytic conductor: \(11.2891\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.06334128894\)
\(L(\frac12)\) \(\approx\) \(0.06334128894\)
\(L(\frac{5}{2})\) 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 \( 1 + 5T \)
good7 \( 1 + 23.9T + 343T^{2} \)
11 \( 1 + 57.9T + 1.33e3T^{2} \)
13 \( 1 + 8.16T + 2.19e3T^{2} \)
17 \( 1 + 50.0T + 4.91e3T^{2} \)
19 \( 1 + 69.7T + 6.85e3T^{2} \)
23 \( 1 + 4.92T + 1.21e4T^{2} \)
29 \( 1 - 79.4T + 2.43e4T^{2} \)
31 \( 1 + 260.T + 2.97e4T^{2} \)
37 \( 1 + 223.T + 5.06e4T^{2} \)
41 \( 1 + 337.T + 6.89e4T^{2} \)
43 \( 1 + 326.T + 7.95e4T^{2} \)
47 \( 1 + 89.6T + 1.03e5T^{2} \)
53 \( 1 + 543.T + 1.48e5T^{2} \)
59 \( 1 + 92T + 2.05e5T^{2} \)
61 \( 1 - 159.T + 2.26e5T^{2} \)
67 \( 1 - 910.T + 3.00e5T^{2} \)
71 \( 1 - 293.T + 3.57e5T^{2} \)
73 \( 1 - 142.T + 3.89e5T^{2} \)
79 \( 1 + 1.10e3T + 4.93e5T^{2} \)
83 \( 1 + 813.T + 5.71e5T^{2} \)
89 \( 1 - 956.T + 7.04e5T^{2} \)
97 \( 1 - 106.T + 9.12e5T^{2} \)
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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

−8.623617526308794824466521541341, −8.034063023759955261150848424966, −7.03314455088647665289434335523, −6.55699466174677142709572415380, −5.50374486558184156591473013830, −4.76160230446448533195050346214, −3.65140508153700652794686853834, −2.95361400622317892568025384428, −1.96972612622166197729846886069, −0.10528843580233186873871118968, 0.10528843580233186873871118968, 1.96972612622166197729846886069, 2.95361400622317892568025384428, 3.65140508153700652794686853834, 4.76160230446448533195050346214, 5.50374486558184156591473013830, 6.55699466174677142709572415380, 7.03314455088647665289434335523, 8.034063023759955261150848424966, 8.623617526308794824466521541341

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