# Properties

 Label 2-15680-1.1-c1-0-52 Degree $2$ Conductor $15680$ Sign $-1$ Analytic cond. $125.205$ Root an. cond. $11.1895$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 − 3-s − 5-s − 2·9-s − 3·11-s + 5·13-s + 15-s − 3·17-s − 2·19-s + 6·23-s + 25-s + 5·27-s − 3·29-s − 4·31-s + 3·33-s − 2·37-s − 5·39-s + 12·41-s − 10·43-s + 2·45-s + 9·47-s + 3·51-s − 12·53-s + 3·55-s + 2·57-s + 8·61-s − 5·65-s − 4·67-s + ⋯
 L(s)  = 1 − 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.904·11-s + 1.38·13-s + 0.258·15-s − 0.727·17-s − 0.458·19-s + 1.25·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s − 0.718·31-s + 0.522·33-s − 0.328·37-s − 0.800·39-s + 1.87·41-s − 1.52·43-s + 0.298·45-s + 1.31·47-s + 0.420·51-s − 1.64·53-s + 0.404·55-s + 0.264·57-s + 1.02·61-s − 0.620·65-s − 0.488·67-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$15680$$    =    $$2^{6} \cdot 5 \cdot 7^{2}$$ Sign: $-1$ Analytic conductor: $$125.205$$ Root analytic conductor: $$11.1895$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{15680} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 15680,\ (\ :1/2),\ -1)$$

## 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$$
7 $$1$$
good3 $$1 + T + p T^{2}$$
11 $$1 + 3 T + p T^{2}$$
13 $$1 - 5 T + p T^{2}$$
17 $$1 + 3 T + p T^{2}$$
19 $$1 + 2 T + p T^{2}$$
23 $$1 - 6 T + p T^{2}$$
29 $$1 + 3 T + p T^{2}$$
31 $$1 + 4 T + p T^{2}$$
37 $$1 + 2 T + p T^{2}$$
41 $$1 - 12 T + p T^{2}$$
43 $$1 + 10 T + p T^{2}$$
47 $$1 - 9 T + p T^{2}$$
53 $$1 + 12 T + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 - 8 T + p T^{2}$$
67 $$1 + 4 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 + 2 T + p T^{2}$$
79 $$1 - T + p T^{2}$$
83 $$1 + 12 T + p T^{2}$$
89 $$1 - 12 T + p T^{2}$$
97 $$1 - T + p T^{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

−16.27918058683733, −15.70521205619242, −15.34866029952413, −14.63538749084514, −14.08799270286763, −13.31053556615522, −12.94368303078648, −12.46612160133975, −11.50298562059002, −11.23528110807813, −10.80675915703285, −10.33180779386039, −9.261472568973643, −8.802161270444983, −8.318985739712541, −7.594941507796575, −6.944351412433016, −6.220785031076169, −5.738494598494589, −5.055805374194028, −4.414894259348672, −3.559915844925244, −2.943367486130850, −2.040492009437095, −0.9144680225949532, 0, 0.9144680225949532, 2.040492009437095, 2.943367486130850, 3.559915844925244, 4.414894259348672, 5.055805374194028, 5.738494598494589, 6.220785031076169, 6.944351412433016, 7.594941507796575, 8.318985739712541, 8.802161270444983, 9.261472568973643, 10.33180779386039, 10.80675915703285, 11.23528110807813, 11.50298562059002, 12.46612160133975, 12.94368303078648, 13.31053556615522, 14.08799270286763, 14.63538749084514, 15.34866029952413, 15.70521205619242, 16.27918058683733