Properties

Label 2-6080-1.1-c1-0-122
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  + 3-s + 5-s − 7-s − 2·9-s + 3·13-s + 15-s − 7·17-s + 19-s − 21-s − 5·23-s + 25-s − 5·27-s + 5·29-s + 10·31-s − 35-s − 2·37-s + 3·39-s + 2·41-s − 6·43-s − 2·45-s − 6·49-s − 7·51-s − 9·53-s + 57-s + 7·59-s + 4·61-s + 2·63-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.832·13-s + 0.258·15-s − 1.69·17-s + 0.229·19-s − 0.218·21-s − 1.04·23-s + 1/5·25-s − 0.962·27-s + 0.928·29-s + 1.79·31-s − 0.169·35-s − 0.328·37-s + 0.480·39-s + 0.312·41-s − 0.914·43-s − 0.298·45-s − 6/7·49-s − 0.980·51-s − 1.23·53-s + 0.132·57-s + 0.911·59-s + 0.512·61-s + 0.251·63-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 \( 1 - T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−7.992163571176218018110340599906, −6.80014773191940009506494915973, −6.37866299170461284822481763215, −5.72906766467486382197758944621, −4.72673976763971238769726906697, −4.00540344967317129351925958514, −3.03876626292382033847060855430, −2.48206108209228295528698972398, −1.47693515273990449433209054058, 0, 1.47693515273990449433209054058, 2.48206108209228295528698972398, 3.03876626292382033847060855430, 4.00540344967317129351925958514, 4.72673976763971238769726906697, 5.72906766467486382197758944621, 6.37866299170461284822481763215, 6.80014773191940009506494915973, 7.992163571176218018110340599906

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