# Properties

 Label 2-6080-1.1-c1-0-126 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 $1$

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## Dirichlet series

 L(s)  = 1 + 2.52·3-s − 5-s − 4.03·7-s + 3.39·9-s + 1.67·11-s + 4.79·13-s − 2.52·15-s − 3.28·17-s − 19-s − 10.2·21-s − 4.03·23-s + 25-s + 0.988·27-s − 1.14·29-s − 9.35·31-s + 4.23·33-s + 4.03·35-s + 4.43·37-s + 12.1·39-s + 3.21·41-s + 4.89·43-s − 3.39·45-s − 6.65·47-s + 9.30·49-s − 8.30·51-s + 0.224·53-s − 1.67·55-s + ⋯
 L(s)  = 1 + 1.45·3-s − 0.447·5-s − 1.52·7-s + 1.13·9-s + 0.505·11-s + 1.32·13-s − 0.652·15-s − 0.796·17-s − 0.229·19-s − 2.22·21-s − 0.841·23-s + 0.200·25-s + 0.190·27-s − 0.212·29-s − 1.67·31-s + 0.737·33-s + 0.682·35-s + 0.728·37-s + 1.94·39-s + 0.502·41-s + 0.746·43-s − 0.505·45-s − 0.970·47-s + 1.32·49-s − 1.16·51-s + 0.0308·53-s − 0.226·55-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: $$1$$ 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 - 2.52T + 3T^{2}$$
7 $$1 + 4.03T + 7T^{2}$$
11 $$1 - 1.67T + 11T^{2}$$
13 $$1 - 4.79T + 13T^{2}$$
17 $$1 + 3.28T + 17T^{2}$$
23 $$1 + 4.03T + 23T^{2}$$
29 $$1 + 1.14T + 29T^{2}$$
31 $$1 + 9.35T + 31T^{2}$$
37 $$1 - 4.43T + 37T^{2}$$
41 $$1 - 3.21T + 41T^{2}$$
43 $$1 - 4.89T + 43T^{2}$$
47 $$1 + 6.65T + 47T^{2}$$
53 $$1 - 0.224T + 53T^{2}$$
59 $$1 - 5.52T + 59T^{2}$$
61 $$1 + 12.9T + 61T^{2}$$
67 $$1 + 11.5T + 67T^{2}$$
71 $$1 + 1.21T + 71T^{2}$$
73 $$1 + 4.98T + 73T^{2}$$
79 $$1 + 6.00T + 79T^{2}$$
83 $$1 + 10.6T + 83T^{2}$$
89 $$1 + 4.37T + 89T^{2}$$
97 $$1 - 1.93T + 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

−7.72439857712141411292403630548, −7.19239376433992938585277680572, −6.29675972334452097881277046879, −5.90136256668388925194816398408, −4.36008552596490965105025896843, −3.78955106248052868805425185798, −3.32576948442519179525523618890, −2.53561718008445474249134133680, −1.53371405299089746699836874318, 0, 1.53371405299089746699836874318, 2.53561718008445474249134133680, 3.32576948442519179525523618890, 3.78955106248052868805425185798, 4.36008552596490965105025896843, 5.90136256668388925194816398408, 6.29675972334452097881277046879, 7.19239376433992938585277680572, 7.72439857712141411292403630548