Properties

Label 2-2160-1.1-c3-0-94
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 yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·5-s + 30.2·7-s + 67.3·11-s − 60.7·13-s − 61.1·17-s + 27.8·19-s − 40.4·23-s + 25·25-s − 212.·29-s − 167.·31-s − 151.·35-s − 366.·37-s + 363.·41-s − 153.·43-s − 434.·47-s + 570.·49-s − 79.6·53-s − 336.·55-s + 339.·59-s − 525.·61-s + 303.·65-s − 131.·67-s − 296.·71-s − 1.23e3·73-s + 2.03e3·77-s + 621.·79-s − 76.3·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.63·7-s + 1.84·11-s − 1.29·13-s − 0.872·17-s + 0.336·19-s − 0.366·23-s + 0.200·25-s − 1.35·29-s − 0.969·31-s − 0.729·35-s − 1.62·37-s + 1.38·41-s − 0.545·43-s − 1.34·47-s + 1.66·49-s − 0.206·53-s − 0.825·55-s + 0.748·59-s − 1.10·61-s + 0.580·65-s − 0.240·67-s − 0.495·71-s − 1.97·73-s + 3.01·77-s + 0.885·79-s − 0.100·83-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: no
Arithmetic: yes
Character: Trivial
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 \( 1 + 5T \)
good7 \( 1 - 30.2T + 343T^{2} \)
11 \( 1 - 67.3T + 1.33e3T^{2} \)
13 \( 1 + 60.7T + 2.19e3T^{2} \)
17 \( 1 + 61.1T + 4.91e3T^{2} \)
19 \( 1 - 27.8T + 6.85e3T^{2} \)
23 \( 1 + 40.4T + 1.21e4T^{2} \)
29 \( 1 + 212.T + 2.43e4T^{2} \)
31 \( 1 + 167.T + 2.97e4T^{2} \)
37 \( 1 + 366.T + 5.06e4T^{2} \)
41 \( 1 - 363.T + 6.89e4T^{2} \)
43 \( 1 + 153.T + 7.95e4T^{2} \)
47 \( 1 + 434.T + 1.03e5T^{2} \)
53 \( 1 + 79.6T + 1.48e5T^{2} \)
59 \( 1 - 339.T + 2.05e5T^{2} \)
61 \( 1 + 525.T + 2.26e5T^{2} \)
67 \( 1 + 131.T + 3.00e5T^{2} \)
71 \( 1 + 296.T + 3.57e5T^{2} \)
73 \( 1 + 1.23e3T + 3.89e5T^{2} \)
79 \( 1 - 621.T + 4.93e5T^{2} \)
83 \( 1 + 76.3T + 5.71e5T^{2} \)
89 \( 1 - 192.T + 7.04e5T^{2} \)
97 \( 1 + 874.T + 9.12e5T^{2} \)
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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

−8.384127679661882645416028014099, −7.42182523151609588089684203761, −7.07347963563550503502391400363, −5.91532471273555059391413711046, −4.93095394823730203375550937906, −4.36496597463423691042810972180, −3.53117768187808471979404951777, −2.05920507445793014945921678138, −1.43910009301941113965131973890, 0, 1.43910009301941113965131973890, 2.05920507445793014945921678138, 3.53117768187808471979404951777, 4.36496597463423691042810972180, 4.93095394823730203375550937906, 5.91532471273555059391413711046, 7.07347963563550503502391400363, 7.42182523151609588089684203761, 8.384127679661882645416028014099

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