Properties

Label 2-2160-1.1-c3-0-86
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 + 7.95·7-s + 37.1·11-s − 35.1·13-s − 99.1·17-s + 44.6·19-s − 102.·23-s + 25·25-s + 285.·29-s − 238.·31-s + 39.7·35-s + 339.·37-s − 423.·41-s − 144.·43-s − 418.·47-s − 279.·49-s + 186.·53-s + 185.·55-s − 293.·59-s − 701.·61-s − 175.·65-s + 292.·67-s − 738.·71-s + 453.·73-s + 295.·77-s − 892.·79-s + 66.4·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.429·7-s + 1.01·11-s − 0.749·13-s − 1.41·17-s + 0.539·19-s − 0.930·23-s + 0.200·25-s + 1.82·29-s − 1.38·31-s + 0.192·35-s + 1.50·37-s − 1.61·41-s − 0.511·43-s − 1.29·47-s − 0.815·49-s + 0.484·53-s + 0.455·55-s − 0.648·59-s − 1.47·61-s − 0.335·65-s + 0.533·67-s − 1.23·71-s + 0.726·73-s + 0.436·77-s − 1.27·79-s + 0.0878·83-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 \( 1 - 7.95T + 343T^{2} \)
11 \( 1 - 37.1T + 1.33e3T^{2} \)
13 \( 1 + 35.1T + 2.19e3T^{2} \)
17 \( 1 + 99.1T + 4.91e3T^{2} \)
19 \( 1 - 44.6T + 6.85e3T^{2} \)
23 \( 1 + 102.T + 1.21e4T^{2} \)
29 \( 1 - 285.T + 2.43e4T^{2} \)
31 \( 1 + 238.T + 2.97e4T^{2} \)
37 \( 1 - 339.T + 5.06e4T^{2} \)
41 \( 1 + 423.T + 6.89e4T^{2} \)
43 \( 1 + 144.T + 7.95e4T^{2} \)
47 \( 1 + 418.T + 1.03e5T^{2} \)
53 \( 1 - 186.T + 1.48e5T^{2} \)
59 \( 1 + 293.T + 2.05e5T^{2} \)
61 \( 1 + 701.T + 2.26e5T^{2} \)
67 \( 1 - 292.T + 3.00e5T^{2} \)
71 \( 1 + 738.T + 3.57e5T^{2} \)
73 \( 1 - 453.T + 3.89e5T^{2} \)
79 \( 1 + 892.T + 4.93e5T^{2} \)
83 \( 1 - 66.4T + 5.71e5T^{2} \)
89 \( 1 - 868.T + 7.04e5T^{2} \)
97 \( 1 - 112.T + 9.12e5T^{2} \)
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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

−8.427199971132498403855049114661, −7.54565259561919279666686949237, −6.65019177973329127314784165489, −6.14535112315621334335229627728, −4.96167930500241211309875740163, −4.44994517994404549184798042231, −3.31258977008535191507790073022, −2.20934440912494615610265893096, −1.39760805951814472794312813056, 0, 1.39760805951814472794312813056, 2.20934440912494615610265893096, 3.31258977008535191507790073022, 4.44994517994404549184798042231, 4.96167930500241211309875740163, 6.14535112315621334335229627728, 6.65019177973329127314784165489, 7.54565259561919279666686949237, 8.427199971132498403855049114661

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