Properties

Label 2-6040-1.1-c1-0-94
Degree $2$
Conductor $6040$
Sign $-1$
Analytic cond. $48.2296$
Root an. cond. $6.94475$
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  − 3-s + 5-s − 7-s − 2·9-s − 11-s − 13-s − 15-s + 2·17-s − 19-s + 21-s + 25-s + 5·27-s + 6·29-s + 33-s − 35-s + 4·37-s + 39-s − 2·41-s − 2·45-s − 6·49-s − 2·51-s + 2·53-s − 55-s + 57-s − 3·59-s + 14·61-s + 2·63-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.377·7-s − 2/3·9-s − 0.301·11-s − 0.277·13-s − 0.258·15-s + 0.485·17-s − 0.229·19-s + 0.218·21-s + 1/5·25-s + 0.962·27-s + 1.11·29-s + 0.174·33-s − 0.169·35-s + 0.657·37-s + 0.160·39-s − 0.312·41-s − 0.298·45-s − 6/7·49-s − 0.280·51-s + 0.274·53-s − 0.134·55-s + 0.132·57-s − 0.390·59-s + 1.79·61-s + 0.251·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6040\)    =    \(2^{3} \cdot 5 \cdot 151\)
Sign: $-1$
Analytic conductor: \(48.2296\)
Root analytic conductor: \(6.94475\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6040,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
151 \( 1 - T \)
good3 \( 1 + T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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

−7.77932244040255403947640205964, −6.75582306573438313665376479569, −6.34302931134054931805233737710, −5.52760641360161750962859992703, −5.07006100942326275278962315301, −4.12496250074832334144634746085, −3.05550757727067772325746999218, −2.46388802151430345528390776161, −1.18107094186407499123959466774, 0, 1.18107094186407499123959466774, 2.46388802151430345528390776161, 3.05550757727067772325746999218, 4.12496250074832334144634746085, 5.07006100942326275278962315301, 5.52760641360161750962859992703, 6.34302931134054931805233737710, 6.75582306573438313665376479569, 7.77932244040255403947640205964

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