# Properties

 Label 2-966-1.1-c1-0-12 Degree $2$ Conductor $966$ Sign $1$ Analytic cond. $7.71354$ Root an. cond. $2.77732$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + 3-s + 4-s + 3.70·5-s − 6-s + 7-s − 8-s + 9-s − 3.70·10-s − 4·11-s + 12-s + 5.70·13-s − 14-s + 3.70·15-s + 16-s + 4·17-s − 18-s − 5.40·19-s + 3.70·20-s + 21-s + 4·22-s + 23-s − 24-s + 8.70·25-s − 5.70·26-s + 27-s + 28-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.65·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.17·10-s − 1.20·11-s + 0.288·12-s + 1.58·13-s − 0.267·14-s + 0.955·15-s + 0.250·16-s + 0.970·17-s − 0.235·18-s − 1.23·19-s + 0.827·20-s + 0.218·21-s + 0.852·22-s + 0.208·23-s − 0.204·24-s + 1.74·25-s − 1.11·26-s + 0.192·27-s + 0.188·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$966$$    =    $$2 \cdot 3 \cdot 7 \cdot 23$$ Sign: $1$ Analytic conductor: $$7.71354$$ Root analytic conductor: $$2.77732$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 966,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.983979540$$ $$L(\frac12)$$ $$\approx$$ $$1.983979540$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 - T$$
7 $$1 - T$$
23 $$1 - T$$
good5 $$1 - 3.70T + 5T^{2}$$
11 $$1 + 4T + 11T^{2}$$
13 $$1 - 5.70T + 13T^{2}$$
17 $$1 - 4T + 17T^{2}$$
19 $$1 + 5.40T + 19T^{2}$$
29 $$1 - 0.298T + 29T^{2}$$
31 $$1 + 2T + 31T^{2}$$
37 $$1 - 4.29T + 37T^{2}$$
41 $$1 + 0.298T + 41T^{2}$$
43 $$1 + 1.70T + 43T^{2}$$
47 $$1 + 11.1T + 47T^{2}$$
53 $$1 + 9.40T + 53T^{2}$$
59 $$1 + 7.40T + 59T^{2}$$
61 $$1 + 2T + 61T^{2}$$
67 $$1 - 14.8T + 67T^{2}$$
71 $$1 + 7.40T + 71T^{2}$$
73 $$1 - 1.40T + 73T^{2}$$
79 $$1 - 8T + 79T^{2}$$
83 $$1 - 13.4T + 83T^{2}$$
89 $$1 + 11.4T + 89T^{2}$$
97 $$1 - 6.29T + 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

−9.953364830239368125571655074213, −9.212867373403171000311331131526, −8.405554987148443698619924414055, −7.83907588814253103422677825385, −6.53075073772413365267487489114, −5.91497445398735978294919446427, −4.94378442004194954455015025915, −3.33061698149894243859735840628, −2.26126253365041427194897828528, −1.39627103285555046757905291301, 1.39627103285555046757905291301, 2.26126253365041427194897828528, 3.33061698149894243859735840628, 4.94378442004194954455015025915, 5.91497445398735978294919446427, 6.53075073772413365267487489114, 7.83907588814253103422677825385, 8.405554987148443698619924414055, 9.212867373403171000311331131526, 9.953364830239368125571655074213