Properties

Label 2-2160-1.1-c1-0-12
Degree $2$
Conductor $2160$
Sign $1$
Analytic cond. $17.2476$
Root an. cond. $4.15303$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 3·7-s − 2·11-s − 5·13-s + 8·17-s − 19-s + 6·23-s + 25-s − 2·29-s + 3·35-s + 5·37-s + 10·41-s − 4·43-s + 4·47-s + 2·49-s + 2·53-s − 2·55-s − 8·59-s + 7·61-s − 5·65-s + 9·67-s + 2·71-s − 5·73-s − 6·77-s + 3·79-s + 6·83-s + 8·85-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.13·7-s − 0.603·11-s − 1.38·13-s + 1.94·17-s − 0.229·19-s + 1.25·23-s + 1/5·25-s − 0.371·29-s + 0.507·35-s + 0.821·37-s + 1.56·41-s − 0.609·43-s + 0.583·47-s + 2/7·49-s + 0.274·53-s − 0.269·55-s − 1.04·59-s + 0.896·61-s − 0.620·65-s + 1.09·67-s + 0.237·71-s − 0.585·73-s − 0.683·77-s + 0.337·79-s + 0.658·83-s + 0.867·85-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: yes
Arithmetic: yes
Character: $\chi_{2160} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.183206545\)
\(L(\frac12)\) \(\approx\) \(2.183206545\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−9.209731645846489900441455328004, −8.033038279448884875142132147382, −7.72379481902030800570658235358, −6.88066201922308126532422526866, −5.60485752225476628583929294992, −5.21952785456145416033046309322, −4.39646975608954166240974959735, −3.07119386896251421193504749200, −2.20929616241085403285528735477, −1.02045420469944533258202986585, 1.02045420469944533258202986585, 2.20929616241085403285528735477, 3.07119386896251421193504749200, 4.39646975608954166240974959735, 5.21952785456145416033046309322, 5.60485752225476628583929294992, 6.88066201922308126532422526866, 7.72379481902030800570658235358, 8.033038279448884875142132147382, 9.209731645846489900441455328004

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