Properties

Label 2-6048-1.1-c1-0-84
Degree $2$
Conductor $6048$
Sign $-1$
Analytic cond. $48.2935$
Root an. cond. $6.94935$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.766·5-s + 7-s + 5.54·11-s + 6.31·13-s − 7.15·17-s − 6.15·19-s − 7.17·23-s − 4.41·25-s − 3.16·29-s − 0.163·31-s − 0.766·35-s + 5.31·37-s − 0.865·41-s − 8.31·43-s − 2.36·47-s + 49-s − 4·53-s − 4.24·55-s − 2.36·59-s − 13.5·61-s − 4.83·65-s + 7.05·67-s − 7.86·71-s − 1.98·73-s + 5.54·77-s + 5.24·79-s + 12.5·83-s + ⋯
L(s)  = 1  − 0.342·5-s + 0.377·7-s + 1.67·11-s + 1.75·13-s − 1.73·17-s − 1.41·19-s − 1.49·23-s − 0.882·25-s − 0.587·29-s − 0.0293·31-s − 0.129·35-s + 0.873·37-s − 0.135·41-s − 1.26·43-s − 0.345·47-s + 0.142·49-s − 0.549·53-s − 0.573·55-s − 0.308·59-s − 1.73·61-s − 0.599·65-s + 0.861·67-s − 0.932·71-s − 0.232·73-s + 0.632·77-s + 0.590·79-s + 1.37·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-1$
Analytic conductor: \(48.2935\)
Root analytic conductor: \(6.94935\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6048,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + 0.766T + 5T^{2} \)
11 \( 1 - 5.54T + 11T^{2} \)
13 \( 1 - 6.31T + 13T^{2} \)
17 \( 1 + 7.15T + 17T^{2} \)
19 \( 1 + 6.15T + 19T^{2} \)
23 \( 1 + 7.17T + 23T^{2} \)
29 \( 1 + 3.16T + 29T^{2} \)
31 \( 1 + 0.163T + 31T^{2} \)
37 \( 1 - 5.31T + 37T^{2} \)
41 \( 1 + 0.865T + 41T^{2} \)
43 \( 1 + 8.31T + 43T^{2} \)
47 \( 1 + 2.36T + 47T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + 2.36T + 59T^{2} \)
61 \( 1 + 13.5T + 61T^{2} \)
67 \( 1 - 7.05T + 67T^{2} \)
71 \( 1 + 7.86T + 71T^{2} \)
73 \( 1 + 1.98T + 73T^{2} \)
79 \( 1 - 5.24T + 79T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 + 4.86T + 89T^{2} \)
97 \( 1 - 3.26T + 97T^{2} \)
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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

−7.964519429135025712330412335563, −6.74404392882548429470555591932, −6.38246172948056386707253847012, −5.84915774576260205627800573054, −4.43271897365286880668708882343, −4.14623802162842325790009360883, −3.50147236565789978357316949051, −2.05885920089271034156037459578, −1.49212911354658975128834941193, 0, 1.49212911354658975128834941193, 2.05885920089271034156037459578, 3.50147236565789978357316949051, 4.14623802162842325790009360883, 4.43271897365286880668708882343, 5.84915774576260205627800573054, 6.38246172948056386707253847012, 6.74404392882548429470555591932, 7.964519429135025712330412335563

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