Properties

Label 2-6080-1.1-c1-0-61
Degree $2$
Conductor $6080$
Sign $-1$
Analytic cond. $48.5490$
Root an. cond. $6.96771$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 5-s + 9-s − 4·11-s − 4·13-s + 2·15-s − 2·17-s + 19-s + 4·23-s + 25-s + 4·27-s + 6·29-s + 8·31-s + 8·33-s + 4·37-s + 8·39-s − 2·41-s + 4·43-s − 45-s + 8·47-s − 7·49-s + 4·51-s + 4·55-s − 2·57-s − 8·59-s − 2·61-s + 4·65-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 1.10·13-s + 0.516·15-s − 0.485·17-s + 0.229·19-s + 0.834·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s + 1.43·31-s + 1.39·33-s + 0.657·37-s + 1.28·39-s − 0.312·41-s + 0.609·43-s − 0.149·45-s + 1.16·47-s − 49-s + 0.560·51-s + 0.539·55-s − 0.264·57-s − 1.04·59-s − 0.256·61-s + 0.496·65-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: yes
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 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−7.68656518057638464254387992403, −6.91696056306923713178838443715, −6.30341525490445618253476785565, −5.48168747503838296966204643422, −4.82015922401866741415673657539, −4.47094060127076576797658975899, −3.03572997981959204678456669366, −2.49396800688193770861386115455, −0.934019685539485854971786712589, 0, 0.934019685539485854971786712589, 2.49396800688193770861386115455, 3.03572997981959204678456669366, 4.47094060127076576797658975899, 4.82015922401866741415673657539, 5.48168747503838296966204643422, 6.30341525490445618253476785565, 6.91696056306923713178838443715, 7.68656518057638464254387992403

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