Properties

Label 2-6080-1.1-c1-0-96
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 5-s + 2·7-s + 9-s − 6·13-s − 2·15-s + 2·17-s + 19-s − 4·21-s − 2·23-s + 25-s + 4·27-s + 2·29-s + 4·31-s + 2·35-s + 10·37-s + 12·39-s − 10·41-s − 6·43-s + 45-s − 6·47-s − 3·49-s − 4·51-s − 6·53-s − 2·57-s + 4·59-s − 2·61-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.66·13-s − 0.516·15-s + 0.485·17-s + 0.229·19-s − 0.872·21-s − 0.417·23-s + 1/5·25-s + 0.769·27-s + 0.371·29-s + 0.718·31-s + 0.338·35-s + 1.64·37-s + 1.92·39-s − 1.56·41-s − 0.914·43-s + 0.149·45-s − 0.875·47-s − 3/7·49-s − 0.560·51-s − 0.824·53-s − 0.264·57-s + 0.520·59-s − 0.256·61-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: yes
Arithmetic: yes
Character: $\chi_{6080} (1, \cdot )$
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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.79497122909070729230975901167, −6.76092766673896308472273583644, −6.35325897786086339440052896451, −5.34581376198822801198219497435, −5.05753034422239968677813934261, −4.42370503190283266986014202128, −3.11484300587933940109704297622, −2.22121268754936937703711757367, −1.19435999771081356370334617186, 0, 1.19435999771081356370334617186, 2.22121268754936937703711757367, 3.11484300587933940109704297622, 4.42370503190283266986014202128, 5.05753034422239968677813934261, 5.34581376198822801198219497435, 6.35325897786086339440052896451, 6.76092766673896308472273583644, 7.79497122909070729230975901167

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