Properties

Label 2-6080-1.1-c1-0-91
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  + 1.44·3-s + 5-s + 2.92·7-s − 0.917·9-s + 4.55·11-s + 1.15·13-s + 1.44·15-s + 5.35·17-s − 19-s + 4.22·21-s + 6.10·23-s + 25-s − 5.65·27-s + 5.09·29-s − 1.54·31-s + 6.57·33-s + 2.92·35-s + 8.92·37-s + 1.66·39-s − 2.88·41-s − 7.86·43-s − 0.917·45-s − 7.58·47-s + 1.55·49-s + 7.73·51-s − 4.48·53-s + 4.55·55-s + ⋯
L(s)  = 1  + 0.833·3-s + 0.447·5-s + 1.10·7-s − 0.305·9-s + 1.37·11-s + 0.320·13-s + 0.372·15-s + 1.29·17-s − 0.229·19-s + 0.921·21-s + 1.27·23-s + 0.200·25-s − 1.08·27-s + 0.945·29-s − 0.276·31-s + 1.14·33-s + 0.494·35-s + 1.46·37-s + 0.266·39-s − 0.450·41-s − 1.19·43-s − 0.136·45-s − 1.10·47-s + 0.222·49-s + 1.08·51-s − 0.615·53-s + 0.614·55-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

\(L(1)\) \(\approx\) \(4.035468609\)
\(L(\frac12)\) \(\approx\) \(4.035468609\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.44T + 3T^{2} \)
7 \( 1 - 2.92T + 7T^{2} \)
11 \( 1 - 4.55T + 11T^{2} \)
13 \( 1 - 1.15T + 13T^{2} \)
17 \( 1 - 5.35T + 17T^{2} \)
23 \( 1 - 6.10T + 23T^{2} \)
29 \( 1 - 5.09T + 29T^{2} \)
31 \( 1 + 1.54T + 31T^{2} \)
37 \( 1 - 8.92T + 37T^{2} \)
41 \( 1 + 2.88T + 41T^{2} \)
43 \( 1 + 7.86T + 43T^{2} \)
47 \( 1 + 7.58T + 47T^{2} \)
53 \( 1 + 4.48T + 53T^{2} \)
59 \( 1 + 2.09T + 59T^{2} \)
61 \( 1 + 1.59T + 61T^{2} \)
67 \( 1 - 3.73T + 67T^{2} \)
71 \( 1 + 3.73T + 71T^{2} \)
73 \( 1 + 9.81T + 73T^{2} \)
79 \( 1 - 4.60T + 79T^{2} \)
83 \( 1 + 9.89T + 83T^{2} \)
89 \( 1 - 1.91T + 89T^{2} \)
97 \( 1 + 15.6T + 97T^{2} \)
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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

−8.240623184427645640348629857674, −7.52815538455152989756805658501, −6.66164030896540386665209228869, −5.97359478367925868237506953614, −5.13507147670734243808151924908, −4.43651776910465323165503638201, −3.46570836443418083716668959372, −2.87611055352162256480396468265, −1.73686685906974836148517358823, −1.15482233636226196929798596924, 1.15482233636226196929798596924, 1.73686685906974836148517358823, 2.87611055352162256480396468265, 3.46570836443418083716668959372, 4.43651776910465323165503638201, 5.13507147670734243808151924908, 5.97359478367925868237506953614, 6.66164030896540386665209228869, 7.52815538455152989756805658501, 8.240623184427645640348629857674

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