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Properties

Label 2-2160-12.11-c3-0-46
Degree $2$
Conductor $2160$
Sign $0.866 - 0.5i$
Analytic cond. $127.444$
Root an. cond. $11.2891$
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  + 5i·5-s + 6.39i·7-s − 55.5·11-s + 66.4·13-s − 5.75i·17-s + 20.4i·19-s − 23.9·23-s − 25·25-s − 264. i·29-s − 54.2i·31-s − 31.9·35-s + 273.·37-s − 169. i·41-s + 180. i·43-s − 124.·47-s + ⋯
L(s)  = 1  + 0.447i·5-s + 0.345i·7-s − 1.52·11-s + 1.41·13-s − 0.0821i·17-s + 0.247i·19-s − 0.216·23-s − 0.200·25-s − 1.69i·29-s − 0.314i·31-s − 0.154·35-s + 1.21·37-s − 0.646i·41-s + 0.641i·43-s − 0.385·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.866 - 0.5i$
Analytic conductor: \(127.444\)
Root analytic conductor: \(11.2891\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :3/2),\ 0.866 - 0.5i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.890260022\)
\(L(\frac12)\) \(\approx\) \(1.890260022\)
\(L(\frac{5}{2})\) 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 \( 1 - 5iT \)
good7 \( 1 - 6.39iT - 343T^{2} \)
11 \( 1 + 55.5T + 1.33e3T^{2} \)
13 \( 1 - 66.4T + 2.19e3T^{2} \)
17 \( 1 + 5.75iT - 4.91e3T^{2} \)
19 \( 1 - 20.4iT - 6.85e3T^{2} \)
23 \( 1 + 23.9T + 1.21e4T^{2} \)
29 \( 1 + 264. iT - 2.43e4T^{2} \)
31 \( 1 + 54.2iT - 2.97e4T^{2} \)
37 \( 1 - 273.T + 5.06e4T^{2} \)
41 \( 1 + 169. iT - 6.89e4T^{2} \)
43 \( 1 - 180. iT - 7.95e4T^{2} \)
47 \( 1 + 124.T + 1.03e5T^{2} \)
53 \( 1 - 174. iT - 1.48e5T^{2} \)
59 \( 1 + 19.6T + 2.05e5T^{2} \)
61 \( 1 + 676.T + 2.26e5T^{2} \)
67 \( 1 - 374. iT - 3.00e5T^{2} \)
71 \( 1 - 976.T + 3.57e5T^{2} \)
73 \( 1 + 246.T + 3.89e5T^{2} \)
79 \( 1 + 1.15e3iT - 4.93e5T^{2} \)
83 \( 1 - 59.4T + 5.71e5T^{2} \)
89 \( 1 - 474. iT - 7.04e5T^{2} \)
97 \( 1 + 54.9T + 9.12e5T^{2} \)
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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

−8.669519512900225061282003980356, −7.964392301632177876210372413616, −7.43299966936985239172252599396, −6.10447125811789049664705295830, −5.93407712051969950145809085038, −4.77977783981468522143329409372, −3.82641453147429050483143799988, −2.85878917030116314086114523600, −2.07723384286559144756813924619, −0.66382701093258941288910862883, 0.58672767959591953008119630480, 1.61357106805049579489578533181, 2.83556649983224501775013433541, 3.71262903989269404234624107939, 4.71694668492310038513062788324, 5.43193573768728249115709418948, 6.24049869292250013225880596259, 7.18291889445321038064129568238, 8.008788808974408410188414232833, 8.545339376661760111527209961088

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