# Properties

 Label 2-960-1.1-c3-0-4 Degree $2$ Conductor $960$ Sign $1$ Analytic cond. $56.6418$ Root an. cond. $7.52607$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 3·3-s + 5·5-s − 20·7-s + 9·9-s − 56·11-s + 86·13-s − 15·15-s − 106·17-s + 4·19-s + 60·21-s − 136·23-s + 25·25-s − 27·27-s + 206·29-s + 152·31-s + 168·33-s − 100·35-s − 282·37-s − 258·39-s − 246·41-s + 412·43-s + 45·45-s − 40·47-s + 57·49-s + 318·51-s + 126·53-s − 280·55-s + ⋯
 L(s)  = 1 − 0.577·3-s + 0.447·5-s − 1.07·7-s + 1/3·9-s − 1.53·11-s + 1.83·13-s − 0.258·15-s − 1.51·17-s + 0.0482·19-s + 0.623·21-s − 1.23·23-s + 1/5·25-s − 0.192·27-s + 1.31·29-s + 0.880·31-s + 0.886·33-s − 0.482·35-s − 1.25·37-s − 1.05·39-s − 0.937·41-s + 1.46·43-s + 0.149·45-s − 0.124·47-s + 0.166·49-s + 0.873·51-s + 0.326·53-s − 0.686·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$960$$    =    $$2^{6} \cdot 3 \cdot 5$$ Sign: $1$ Analytic conductor: $$56.6418$$ Root analytic conductor: $$7.52607$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 960,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.119263626$$ $$L(\frac12)$$ $$\approx$$ $$1.119263626$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + p T$$
5 $$1 - p T$$
good7 $$1 + 20 T + p^{3} T^{2}$$
11 $$1 + 56 T + p^{3} T^{2}$$
13 $$1 - 86 T + p^{3} T^{2}$$
17 $$1 + 106 T + p^{3} T^{2}$$
19 $$1 - 4 T + p^{3} T^{2}$$
23 $$1 + 136 T + p^{3} T^{2}$$
29 $$1 - 206 T + p^{3} T^{2}$$
31 $$1 - 152 T + p^{3} T^{2}$$
37 $$1 + 282 T + p^{3} T^{2}$$
41 $$1 + 6 p T + p^{3} T^{2}$$
43 $$1 - 412 T + p^{3} T^{2}$$
47 $$1 + 40 T + p^{3} T^{2}$$
53 $$1 - 126 T + p^{3} T^{2}$$
59 $$1 - 56 T + p^{3} T^{2}$$
61 $$1 - 2 T + p^{3} T^{2}$$
67 $$1 + 388 T + p^{3} T^{2}$$
71 $$1 - 672 T + p^{3} T^{2}$$
73 $$1 - 1170 T + p^{3} T^{2}$$
79 $$1 + 408 T + p^{3} T^{2}$$
83 $$1 - 668 T + p^{3} T^{2}$$
89 $$1 - 66 T + p^{3} T^{2}$$
97 $$1 + 926 T + p^{3} 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

−9.840165869656113900242108330679, −8.780433855590852940848646976830, −8.097421114944634230891736022617, −6.73866858888088326338016793969, −6.27949137795096327554088179549, −5.47290449320686859309662800371, −4.37106354848871932883604818187, −3.24188000628599877559073452225, −2.11620317809945204600237502109, −0.56486487503757291429927248403, 0.56486487503757291429927248403, 2.11620317809945204600237502109, 3.24188000628599877559073452225, 4.37106354848871932883604818187, 5.47290449320686859309662800371, 6.27949137795096327554088179549, 6.73866858888088326338016793969, 8.097421114944634230891736022617, 8.780433855590852940848646976830, 9.840165869656113900242108330679