Properties

Label 2-2160-1.1-c3-0-57
Degree $2$
Conductor $2160$
Sign $-1$
Analytic cond. $127.444$
Root an. cond. $11.2891$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·5-s − 14·7-s + 3·11-s + 47·13-s + 39·17-s − 32·19-s − 99·23-s + 25·25-s − 51·29-s − 83·31-s + 70·35-s + 314·37-s + 108·41-s − 299·43-s + 531·47-s − 147·49-s − 564·53-s − 15·55-s + 12·59-s + 230·61-s − 235·65-s + 268·67-s + 120·71-s + 1.10e3·73-s − 42·77-s + 739·79-s + 1.08e3·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s + 0.0822·11-s + 1.00·13-s + 0.556·17-s − 0.386·19-s − 0.897·23-s + 1/5·25-s − 0.326·29-s − 0.480·31-s + 0.338·35-s + 1.39·37-s + 0.411·41-s − 1.06·43-s + 1.64·47-s − 3/7·49-s − 1.46·53-s − 0.0367·55-s + 0.0264·59-s + 0.482·61-s − 0.448·65-s + 0.488·67-s + 0.200·71-s + 1.77·73-s − 0.0621·77-s + 1.05·79-s + 1.43·83-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: yes
Arithmetic: yes
Character: $\chi_{2160} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2160,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
good7 \( 1 + 2 p T + p^{3} T^{2} \)
11 \( 1 - 3 T + p^{3} T^{2} \)
13 \( 1 - 47 T + p^{3} T^{2} \)
17 \( 1 - 39 T + p^{3} T^{2} \)
19 \( 1 + 32 T + p^{3} T^{2} \)
23 \( 1 + 99 T + p^{3} T^{2} \)
29 \( 1 + 51 T + p^{3} T^{2} \)
31 \( 1 + 83 T + p^{3} T^{2} \)
37 \( 1 - 314 T + p^{3} T^{2} \)
41 \( 1 - 108 T + p^{3} T^{2} \)
43 \( 1 + 299 T + p^{3} T^{2} \)
47 \( 1 - 531 T + p^{3} T^{2} \)
53 \( 1 + 564 T + p^{3} T^{2} \)
59 \( 1 - 12 T + p^{3} T^{2} \)
61 \( 1 - 230 T + p^{3} T^{2} \)
67 \( 1 - 4 p T + p^{3} T^{2} \)
71 \( 1 - 120 T + p^{3} T^{2} \)
73 \( 1 - 1106 T + p^{3} T^{2} \)
79 \( 1 - 739 T + p^{3} T^{2} \)
83 \( 1 - 1086 T + p^{3} T^{2} \)
89 \( 1 - 120 T + p^{3} T^{2} \)
97 \( 1 + 1642 T + p^{3} T^{2} \)
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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

−8.206039674499400801554573790353, −7.72142226606806063044670802921, −6.61529085269633912618643299202, −6.13453375264893980508031804893, −5.18104058540511342490114445103, −4.00189895270115980673101165090, −3.53443410936500912091243470426, −2.41384144506462781980835384211, −1.13587995431193979124584298365, 0, 1.13587995431193979124584298365, 2.41384144506462781980835384211, 3.53443410936500912091243470426, 4.00189895270115980673101165090, 5.18104058540511342490114445103, 6.13453375264893980508031804893, 6.61529085269633912618643299202, 7.72142226606806063044670802921, 8.206039674499400801554573790353

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