Properties

 Degree $2$ Conductor $128$ Sign $1$ Motivic weight $3$ Primitive yes Self-dual yes Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 4.92·3-s − 15.8·5-s + 17.8·7-s − 2.71·9-s + 52.9·11-s − 8.43·13-s + 78.1·15-s + 129.·17-s + 50.4·19-s − 87.9·21-s − 128.·23-s + 126.·25-s + 146.·27-s − 111.·29-s + 302.·31-s − 260.·33-s − 283.·35-s + 182.·37-s + 41.5·39-s − 94.5·41-s + 184.·43-s + 43.0·45-s − 296.·47-s − 24.1·49-s − 636.·51-s + 102.·53-s − 839.·55-s + ⋯
 L(s)  = 1 − 0.948·3-s − 1.41·5-s + 0.964·7-s − 0.100·9-s + 1.45·11-s − 0.179·13-s + 1.34·15-s + 1.84·17-s + 0.609·19-s − 0.914·21-s − 1.16·23-s + 1.01·25-s + 1.04·27-s − 0.712·29-s + 1.75·31-s − 1.37·33-s − 1.36·35-s + 0.813·37-s + 0.170·39-s − 0.360·41-s + 0.654·43-s + 0.142·45-s − 0.921·47-s − 0.0704·49-s − 1.74·51-s + 0.266·53-s − 2.05·55-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$128$$    =    $$2^{7}$$ Sign: $1$ Motivic weight: $$3$$ Character: $\chi_{128} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 128,\ (\ :3/2),\ 1)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$1.02101$$ $$L(\frac12)$$ $$\approx$$ $$1.02101$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
good3 $$1 + 4.92T + 27T^{2}$$
5 $$1 + 15.8T + 125T^{2}$$
7 $$1 - 17.8T + 343T^{2}$$
11 $$1 - 52.9T + 1.33e3T^{2}$$
13 $$1 + 8.43T + 2.19e3T^{2}$$
17 $$1 - 129.T + 4.91e3T^{2}$$
19 $$1 - 50.4T + 6.85e3T^{2}$$
23 $$1 + 128.T + 1.21e4T^{2}$$
29 $$1 + 111.T + 2.43e4T^{2}$$
31 $$1 - 302.T + 2.97e4T^{2}$$
37 $$1 - 182.T + 5.06e4T^{2}$$
41 $$1 + 94.5T + 6.89e4T^{2}$$
43 $$1 - 184.T + 7.95e4T^{2}$$
47 $$1 + 296.T + 1.03e5T^{2}$$
53 $$1 - 102.T + 1.48e5T^{2}$$
59 $$1 + 93.3T + 2.05e5T^{2}$$
61 $$1 - 338.T + 2.26e5T^{2}$$
67 $$1 - 489.T + 3.00e5T^{2}$$
71 $$1 - 86.9T + 3.57e5T^{2}$$
73 $$1 + 154.T + 3.89e5T^{2}$$
79 $$1 - 449.T + 4.93e5T^{2}$$
83 $$1 - 383.T + 5.71e5T^{2}$$
89 $$1 + 517.T + 7.04e5T^{2}$$
97 $$1 - 1.73e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$