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Properties

Label 2-2160-12.11-c3-0-11
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 no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5i·5-s + 20.7i·7-s + 36.3·11-s − 47·13-s − 21i·17-s − 62.3i·19-s + 36.3·23-s − 25·25-s + 123i·29-s + 25.9i·31-s − 103.·35-s − 178·37-s + 342i·41-s + 233. i·43-s + 306.·47-s + ⋯
L(s)  = 1  + 0.447i·5-s + 1.12i·7-s + 0.996·11-s − 1.00·13-s − 0.299i·17-s − 0.752i·19-s + 0.329·23-s − 0.200·25-s + 0.787i·29-s + 0.150i·31-s − 0.501·35-s − 0.790·37-s + 1.30i·41-s + 0.829i·43-s + 0.951·47-s + ⋯

Functional equation

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

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: $\chi_{2160} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8901127041\)
\(L(\frac12)\) \(\approx\) \(0.8901127041\)
\(L(\frac{5}{2})\) 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 \( 1 - 20.7iT - 343T^{2} \)
11 \( 1 - 36.3T + 1.33e3T^{2} \)
13 \( 1 + 47T + 2.19e3T^{2} \)
17 \( 1 + 21iT - 4.91e3T^{2} \)
19 \( 1 + 62.3iT - 6.85e3T^{2} \)
23 \( 1 - 36.3T + 1.21e4T^{2} \)
29 \( 1 - 123iT - 2.43e4T^{2} \)
31 \( 1 - 25.9iT - 2.97e4T^{2} \)
37 \( 1 + 178T + 5.06e4T^{2} \)
41 \( 1 - 342iT - 6.89e4T^{2} \)
43 \( 1 - 233. iT - 7.95e4T^{2} \)
47 \( 1 - 306.T + 1.03e5T^{2} \)
53 \( 1 - 414iT - 1.48e5T^{2} \)
59 \( 1 + 446.T + 2.05e5T^{2} \)
61 \( 1 - 542T + 2.26e5T^{2} \)
67 \( 1 - 155. iT - 3.00e5T^{2} \)
71 \( 1 + 852.T + 3.57e5T^{2} \)
73 \( 1 - 232T + 3.89e5T^{2} \)
79 \( 1 - 348. iT - 4.93e5T^{2} \)
83 \( 1 + 405.T + 5.71e5T^{2} \)
89 \( 1 + 1.35e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.04e3T + 9.12e5T^{2} \)
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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

−9.148701862920846544285939254152, −8.496245740881756200056824224534, −7.40629457137832818037287898564, −6.82074837197660996824644662399, −5.99791061500127688941836083757, −5.15295650009407186980454866880, −4.35056165086576368470179048200, −3.10462251469429307071810747416, −2.50049631462422821294937239586, −1.33735204124623489846335512415, 0.18512213008751159904039247649, 1.17473480166577947928305689713, 2.19690992331681416789362765604, 3.63908584709686883285952056317, 4.13223003103368139136883782474, 5.05968999715628395222260700937, 5.97381611044847364163438242546, 6.95800716291271090520981687587, 7.43107958000274690565680051857, 8.349763197766835549413694720404

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