Properties

Label 2-6080-1.1-c1-0-46
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.56·3-s + 5-s + 0.438·7-s − 0.561·9-s + 4·11-s + 1.56·13-s − 1.56·15-s + 3.56·17-s + 19-s − 0.684·21-s + 6.68·23-s + 25-s + 5.56·27-s + 7.56·29-s + 3.12·31-s − 6.24·33-s + 0.438·35-s − 7.12·37-s − 2.43·39-s − 8.24·41-s + 2·43-s − 0.561·45-s − 5.12·47-s − 6.80·49-s − 5.56·51-s − 2.43·53-s + 4·55-s + ⋯
L(s)  = 1  − 0.901·3-s + 0.447·5-s + 0.165·7-s − 0.187·9-s + 1.20·11-s + 0.433·13-s − 0.403·15-s + 0.863·17-s + 0.229·19-s − 0.149·21-s + 1.39·23-s + 0.200·25-s + 1.07·27-s + 1.40·29-s + 0.560·31-s − 1.08·33-s + 0.0741·35-s − 1.17·37-s − 0.390·39-s − 1.28·41-s + 0.304·43-s − 0.0837·45-s − 0.747·47-s − 0.972·49-s − 0.778·51-s − 0.334·53-s + 0.539·55-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: \(0\)
Selberg data: \((2,\ 6080,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.887819397\)
\(L(\frac12)\) \(\approx\) \(1.887819397\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.56T + 3T^{2} \)
7 \( 1 - 0.438T + 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 - 1.56T + 13T^{2} \)
17 \( 1 - 3.56T + 17T^{2} \)
23 \( 1 - 6.68T + 23T^{2} \)
29 \( 1 - 7.56T + 29T^{2} \)
31 \( 1 - 3.12T + 31T^{2} \)
37 \( 1 + 7.12T + 37T^{2} \)
41 \( 1 + 8.24T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + 5.12T + 47T^{2} \)
53 \( 1 + 2.43T + 53T^{2} \)
59 \( 1 - 7.80T + 59T^{2} \)
61 \( 1 - 13.1T + 61T^{2} \)
67 \( 1 - 6.43T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 5.80T + 73T^{2} \)
79 \( 1 + 3.12T + 79T^{2} \)
83 \( 1 - 14T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 7.12T + 97T^{2} \)
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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

−8.315507256576916796396892464541, −6.94535523318775817339876394312, −6.70848009831705645681524931654, −5.92937235678922846082190471563, −5.23427759648062209665367516142, −4.70884087838282243015241458265, −3.59370650440738886990955010702, −2.88509775242745812307144029991, −1.53306908255023097746391943587, −0.832187340095773646565405038223, 0.832187340095773646565405038223, 1.53306908255023097746391943587, 2.88509775242745812307144029991, 3.59370650440738886990955010702, 4.70884087838282243015241458265, 5.23427759648062209665367516142, 5.92937235678922846082190471563, 6.70848009831705645681524931654, 6.94535523318775817339876394312, 8.315507256576916796396892464541

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