Properties

Label 2-2160-1.1-c3-0-13
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s − 17·7-s + 30·11-s − 61·13-s − 120·17-s + 43·19-s − 90·23-s + 25·25-s − 90·29-s − 8·31-s − 85·35-s + 317·37-s − 30·41-s + 220·43-s + 180·47-s − 54·49-s − 630·53-s + 150·55-s + 840·59-s + 599·61-s − 305·65-s − 107·67-s + 210·71-s − 421·73-s − 510·77-s − 353·79-s + 1.35e3·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.917·7-s + 0.822·11-s − 1.30·13-s − 1.71·17-s + 0.519·19-s − 0.815·23-s + 1/5·25-s − 0.576·29-s − 0.0463·31-s − 0.410·35-s + 1.40·37-s − 0.114·41-s + 0.780·43-s + 0.558·47-s − 0.157·49-s − 1.63·53-s + 0.367·55-s + 1.85·59-s + 1.25·61-s − 0.582·65-s − 0.195·67-s + 0.351·71-s − 0.674·73-s − 0.754·77-s − 0.502·79-s + 1.78·83-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: \(0\)
Selberg data: \((2,\ 2160,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.469750079\)
\(L(\frac12)\) \(\approx\) \(1.469750079\)
\(L(\frac{5}{2})\) 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 \( 1 + 17 T + p^{3} T^{2} \)
11 \( 1 - 30 T + p^{3} T^{2} \)
13 \( 1 + 61 T + p^{3} T^{2} \)
17 \( 1 + 120 T + p^{3} T^{2} \)
19 \( 1 - 43 T + p^{3} T^{2} \)
23 \( 1 + 90 T + p^{3} T^{2} \)
29 \( 1 + 90 T + p^{3} T^{2} \)
31 \( 1 + 8 T + p^{3} T^{2} \)
37 \( 1 - 317 T + p^{3} T^{2} \)
41 \( 1 + 30 T + p^{3} T^{2} \)
43 \( 1 - 220 T + p^{3} T^{2} \)
47 \( 1 - 180 T + p^{3} T^{2} \)
53 \( 1 + 630 T + p^{3} T^{2} \)
59 \( 1 - 840 T + p^{3} T^{2} \)
61 \( 1 - 599 T + p^{3} T^{2} \)
67 \( 1 + 107 T + p^{3} T^{2} \)
71 \( 1 - 210 T + p^{3} T^{2} \)
73 \( 1 + 421 T + p^{3} T^{2} \)
79 \( 1 + 353 T + p^{3} T^{2} \)
83 \( 1 - 1350 T + p^{3} T^{2} \)
89 \( 1 - 1020 T + p^{3} T^{2} \)
97 \( 1 + 997 T + p^{3} T^{2} \)
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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

−8.968483870366647127048734193145, −7.899492181163311703589294261157, −6.98322401260700648544472296795, −6.47295268971550636881682568625, −5.66722375234142070471035674916, −4.63630199511875558860635317980, −3.86447878562413846919079471243, −2.72034048943192975053245375747, −1.97725404593012011966472774398, −0.52544819686275328985980179722, 0.52544819686275328985980179722, 1.97725404593012011966472774398, 2.72034048943192975053245375747, 3.86447878562413846919079471243, 4.63630199511875558860635317980, 5.66722375234142070471035674916, 6.47295268971550636881682568625, 6.98322401260700648544472296795, 7.899492181163311703589294261157, 8.968483870366647127048734193145

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