Properties

 Label 2-3200-1.1-c1-0-3 Degree $2$ Conductor $3200$ Sign $1$ Analytic cond. $25.5521$ Root an. cond. $5.05491$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 1.70·3-s − 2.63·7-s − 0.0783·9-s − 5.41·11-s + 6.34·13-s − 3.41·17-s − 3.26·19-s + 4.49·21-s − 1.36·23-s + 5.26·27-s − 2·29-s − 4.68·31-s + 9.26·33-s − 5.75·37-s − 10.8·39-s − 7.75·41-s + 4.44·43-s − 4.78·47-s − 0.0783·49-s + 5.84·51-s + 1.65·53-s + 5.57·57-s + 3.26·59-s + 2.49·61-s + 0.206·63-s + 7.86·67-s + 2.34·69-s + ⋯
 L(s)  = 1 − 0.986·3-s − 0.994·7-s − 0.0261·9-s − 1.63·11-s + 1.75·13-s − 0.829·17-s − 0.748·19-s + 0.981·21-s − 0.285·23-s + 1.01·27-s − 0.371·29-s − 0.840·31-s + 1.61·33-s − 0.946·37-s − 1.73·39-s − 1.21·41-s + 0.678·43-s − 0.698·47-s − 0.0111·49-s + 0.818·51-s + 0.227·53-s + 0.738·57-s + 0.424·59-s + 0.319·61-s + 0.0259·63-s + 0.960·67-s + 0.281·69-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$3200$$    =    $$2^{7} \cdot 5^{2}$$ Sign: $1$ Analytic conductor: $$25.5521$$ Root analytic conductor: $$5.05491$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{3200} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 3200,\ (\ :1/2),\ 1)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.4957691566$$ $$L(\frac12)$$ $$\approx$$ $$0.4957691566$$ $$L(\frac{3}{2})$$ 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$$
good3 $$1 + 1.70T + 3T^{2}$$
7 $$1 + 2.63T + 7T^{2}$$
11 $$1 + 5.41T + 11T^{2}$$
13 $$1 - 6.34T + 13T^{2}$$
17 $$1 + 3.41T + 17T^{2}$$
19 $$1 + 3.26T + 19T^{2}$$
23 $$1 + 1.36T + 23T^{2}$$
29 $$1 + 2T + 29T^{2}$$
31 $$1 + 4.68T + 31T^{2}$$
37 $$1 + 5.75T + 37T^{2}$$
41 $$1 + 7.75T + 41T^{2}$$
43 $$1 - 4.44T + 43T^{2}$$
47 $$1 + 4.78T + 47T^{2}$$
53 $$1 - 1.65T + 53T^{2}$$
59 $$1 - 3.26T + 59T^{2}$$
61 $$1 - 2.49T + 61T^{2}$$
67 $$1 - 7.86T + 67T^{2}$$
71 $$1 + 6.15T + 71T^{2}$$
73 $$1 - 13.5T + 73T^{2}$$
79 $$1 - 12.6T + 79T^{2}$$
83 $$1 - 14.9T + 83T^{2}$$
89 $$1 + 8.52T + 89T^{2}$$
97 $$1 - 4.58T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$