Properties

Label 2-2160-1.1-c1-0-13
Degree $2$
Conductor $2160$
Sign $1$
Analytic cond. $17.2476$
Root an. cond. $4.15303$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 2.60·7-s − 4.60·11-s + 6.60·13-s + 1.60·17-s − 3.60·19-s + 3·23-s + 25-s + 1.39·29-s + 5.60·31-s + 2.60·35-s + 2·37-s − 4.60·41-s − 0.605·43-s + 9.21·47-s − 0.211·49-s + 1.60·53-s − 4.60·55-s + 1.39·59-s − 4.21·61-s + 6.60·65-s + 0.788·67-s − 7.39·71-s + 12.6·73-s − 12·77-s + 11.6·79-s − 3·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.984·7-s − 1.38·11-s + 1.83·13-s + 0.389·17-s − 0.827·19-s + 0.625·23-s + 0.200·25-s + 0.258·29-s + 1.00·31-s + 0.440·35-s + 0.328·37-s − 0.719·41-s − 0.0923·43-s + 1.34·47-s − 0.0301·49-s + 0.220·53-s − 0.621·55-s + 0.181·59-s − 0.539·61-s + 0.819·65-s + 0.0963·67-s − 0.877·71-s + 1.47·73-s − 1.36·77-s + 1.30·79-s − 0.329·83-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $1$
Analytic conductor: \(17.2476\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.238869039\)
\(L(\frac12)\) \(\approx\) \(2.238869039\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good7 \( 1 - 2.60T + 7T^{2} \)
11 \( 1 + 4.60T + 11T^{2} \)
13 \( 1 - 6.60T + 13T^{2} \)
17 \( 1 - 1.60T + 17T^{2} \)
19 \( 1 + 3.60T + 19T^{2} \)
23 \( 1 - 3T + 23T^{2} \)
29 \( 1 - 1.39T + 29T^{2} \)
31 \( 1 - 5.60T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 4.60T + 41T^{2} \)
43 \( 1 + 0.605T + 43T^{2} \)
47 \( 1 - 9.21T + 47T^{2} \)
53 \( 1 - 1.60T + 53T^{2} \)
59 \( 1 - 1.39T + 59T^{2} \)
61 \( 1 + 4.21T + 61T^{2} \)
67 \( 1 - 0.788T + 67T^{2} \)
71 \( 1 + 7.39T + 71T^{2} \)
73 \( 1 - 12.6T + 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + 3T + 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−8.837742923231340671809744308312, −8.333270517612557651835064641385, −7.73693540045182477010580738940, −6.64235463329142109884693225609, −5.85374606071339145288905515385, −5.14080040216122721109849735810, −4.30659177047953783058588119882, −3.17082536459215421610774302292, −2.13906968512772871672474254369, −1.04750154385184985082253610685, 1.04750154385184985082253610685, 2.13906968512772871672474254369, 3.17082536459215421610774302292, 4.30659177047953783058588119882, 5.14080040216122721109849735810, 5.85374606071339145288905515385, 6.64235463329142109884693225609, 7.73693540045182477010580738940, 8.333270517612557651835064641385, 8.837742923231340671809744308312

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