Properties

 Label 2-70-1.1-c1-0-0 Degree $2$ Conductor $70$ Sign $1$ Analytic cond. $0.558952$ Root an. cond. $0.747631$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + 2-s + 4-s − 5-s − 7-s + 8-s − 3·9-s − 10-s + 4·11-s − 6·13-s − 14-s + 16-s + 2·17-s − 3·18-s − 20-s + 4·22-s + 25-s − 6·26-s − 28-s + 6·29-s + 8·31-s + 32-s + 2·34-s + 35-s − 3·36-s − 10·37-s − 40-s + 2·41-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 9-s − 0.316·10-s + 1.20·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.707·18-s − 0.223·20-s + 0.852·22-s + 1/5·25-s − 1.17·26-s − 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.342·34-s + 0.169·35-s − 1/2·36-s − 1.64·37-s − 0.158·40-s + 0.312·41-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$70$$    =    $$2 \cdot 5 \cdot 7$$ Sign: $1$ Analytic conductor: $$0.558952$$ Root analytic conductor: $$0.747631$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 70,\ (\ :1/2),\ 1)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$1.180304346$$ $$L(\frac12)$$ $$\approx$$ $$1.180304346$$ $$L(\frac{3}{2})$$ 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$$
5 $$1 + T$$
7 $$1 + T$$
good3 $$1 + p T^{2}$$
11 $$1 - 4 T + p T^{2}$$
13 $$1 + 6 T + p T^{2}$$
17 $$1 - 2 T + p T^{2}$$
19 $$1 + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 - 6 T + p T^{2}$$
31 $$1 - 8 T + p T^{2}$$
37 $$1 + 10 T + p T^{2}$$
41 $$1 - 2 T + p T^{2}$$
43 $$1 - 4 T + p T^{2}$$
47 $$1 - 8 T + p T^{2}$$
53 $$1 + 2 T + p T^{2}$$
59 $$1 + 8 T + p T^{2}$$
61 $$1 + 14 T + p T^{2}$$
67 $$1 + 12 T + p T^{2}$$
71 $$1 + 16 T + p T^{2}$$
73 $$1 - 2 T + p T^{2}$$
79 $$1 + 8 T + p T^{2}$$
83 $$1 - 8 T + p T^{2}$$
89 $$1 - 10 T + p T^{2}$$
97 $$1 - 2 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

−14.53586643730421316812320958185, −13.89417772743578510110013468150, −12.14707369305424687183170289720, −11.96572717133546281866330503535, −10.37599431086570880625729241811, −8.977646092372605556420891081568, −7.45420248729985818641828193491, −6.17616144092315208416665412995, −4.63923341949934829303628265619, −3.00381372509091040158022668795, 3.00381372509091040158022668795, 4.63923341949934829303628265619, 6.17616144092315208416665412995, 7.45420248729985818641828193491, 8.977646092372605556420891081568, 10.37599431086570880625729241811, 11.96572717133546281866330503535, 12.14707369305424687183170289720, 13.89417772743578510110013468150, 14.53586643730421316812320958185