Properties

 Label 2-40e2-1.1-c3-0-107 Degree $2$ Conductor $1600$ Sign $-1$ Analytic cond. $94.4030$ Root an. cond. $9.71612$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

Related objects

Dirichlet series

 L(s)  = 1 + 4.47·3-s + 31.3·7-s − 6.99·9-s + 8.94·11-s − 62·13-s + 46·17-s − 107.·19-s + 140·21-s − 192.·23-s − 152.·27-s + 90·29-s − 152.·31-s + 40.0·33-s − 214·37-s − 277.·39-s − 10·41-s − 67.0·43-s − 398.·47-s + 637.·49-s + 205.·51-s − 678·53-s − 480.·57-s + 411.·59-s − 250·61-s − 219.·63-s + 49.1·67-s − 860·69-s + ⋯
 L(s)  = 1 + 0.860·3-s + 1.69·7-s − 0.259·9-s + 0.245·11-s − 1.32·13-s + 0.656·17-s − 1.29·19-s + 1.45·21-s − 1.74·23-s − 1.08·27-s + 0.576·29-s − 0.880·31-s + 0.211·33-s − 0.950·37-s − 1.13·39-s − 0.0380·41-s − 0.237·43-s − 1.23·47-s + 1.85·49-s + 0.564·51-s − 1.75·53-s − 1.11·57-s + 0.907·59-s − 0.524·61-s − 0.438·63-s + 0.0897·67-s − 1.50·69-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$1600$$    =    $$2^{6} \cdot 5^{2}$$ Sign: $-1$ Analytic conductor: $$94.4030$$ Root analytic conductor: $$9.71612$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{1600} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1600,\ (\ :3/2),\ -1)$$

Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
good3 $$1 - 4.47T + 27T^{2}$$
7 $$1 - 31.3T + 343T^{2}$$
11 $$1 - 8.94T + 1.33e3T^{2}$$
13 $$1 + 62T + 2.19e3T^{2}$$
17 $$1 - 46T + 4.91e3T^{2}$$
19 $$1 + 107.T + 6.85e3T^{2}$$
23 $$1 + 192.T + 1.21e4T^{2}$$
29 $$1 - 90T + 2.43e4T^{2}$$
31 $$1 + 152.T + 2.97e4T^{2}$$
37 $$1 + 214T + 5.06e4T^{2}$$
41 $$1 + 10T + 6.89e4T^{2}$$
43 $$1 + 67.0T + 7.95e4T^{2}$$
47 $$1 + 398.T + 1.03e5T^{2}$$
53 $$1 + 678T + 1.48e5T^{2}$$
59 $$1 - 411.T + 2.05e5T^{2}$$
61 $$1 + 250T + 2.26e5T^{2}$$
67 $$1 - 49.1T + 3.00e5T^{2}$$
71 $$1 + 366.T + 3.57e5T^{2}$$
73 $$1 + 522T + 3.89e5T^{2}$$
79 $$1 - 876.T + 4.93e5T^{2}$$
83 $$1 - 380.T + 5.71e5T^{2}$$
89 $$1 - 970T + 7.04e5T^{2}$$
97 $$1 - 934T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$