Properties

Label 2-6080-1.1-c1-0-58
Degree $2$
Conductor $6080$
Sign $-1$
Analytic cond. $48.5490$
Root an. cond. $6.96771$
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  − 2.32·3-s − 5-s − 1.39·7-s + 2.39·9-s − 4.32·13-s + 2.32·15-s + 0.601·17-s − 19-s + 3.24·21-s + 6.04·23-s + 25-s + 1.39·27-s − 4.60·29-s + 2.79·31-s + 1.39·35-s − 1.07·37-s + 10.0·39-s − 5.44·41-s + 8.64·43-s − 2.39·45-s − 1.85·47-s − 5.04·49-s − 1.39·51-s + 3.11·53-s + 2.32·57-s + 6.69·59-s + 2.64·61-s + ⋯
L(s)  = 1  − 1.34·3-s − 0.447·5-s − 0.528·7-s + 0.799·9-s − 1.19·13-s + 0.599·15-s + 0.145·17-s − 0.229·19-s + 0.708·21-s + 1.26·23-s + 0.200·25-s + 0.269·27-s − 0.854·29-s + 0.502·31-s + 0.236·35-s − 0.176·37-s + 1.60·39-s − 0.850·41-s + 1.31·43-s − 0.357·45-s − 0.269·47-s − 0.720·49-s − 0.195·51-s + 0.428·53-s + 0.307·57-s + 0.871·59-s + 0.338·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6080\)    =    \(2^{6} \cdot 5 \cdot 19\)
Sign: $-1$
Analytic conductor: \(48.5490\)
Root analytic conductor: \(6.96771\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6080,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
19 \( 1 + T \)
good3 \( 1 + 2.32T + 3T^{2} \)
7 \( 1 + 1.39T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4.32T + 13T^{2} \)
17 \( 1 - 0.601T + 17T^{2} \)
23 \( 1 - 6.04T + 23T^{2} \)
29 \( 1 + 4.60T + 29T^{2} \)
31 \( 1 - 2.79T + 31T^{2} \)
37 \( 1 + 1.07T + 37T^{2} \)
41 \( 1 + 5.44T + 41T^{2} \)
43 \( 1 - 8.64T + 43T^{2} \)
47 \( 1 + 1.85T + 47T^{2} \)
53 \( 1 - 3.11T + 53T^{2} \)
59 \( 1 - 6.69T + 59T^{2} \)
61 \( 1 - 2.64T + 61T^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 + 5.59T + 71T^{2} \)
73 \( 1 - 12.6T + 73T^{2} \)
79 \( 1 + 4.64T + 79T^{2} \)
83 \( 1 - 1.20T + 83T^{2} \)
89 \( 1 - 9.44T + 89T^{2} \)
97 \( 1 - 4.51T + 97T^{2} \)
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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

−7.49102318082265447066004934463, −6.87418431848291713880554340574, −6.39199657280680181449048168422, −5.40447213539046233806675738620, −5.06434216971650340156172844528, −4.21443472623390800840213966201, −3.28174272213962729051488907582, −2.33204272877008618830183645830, −0.919523848188767636386157686844, 0, 0.919523848188767636386157686844, 2.33204272877008618830183645830, 3.28174272213962729051488907582, 4.21443472623390800840213966201, 5.06434216971650340156172844528, 5.40447213539046233806675738620, 6.39199657280680181449048168422, 6.87418431848291713880554340574, 7.49102318082265447066004934463

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