Properties

 Label 2-700-5.4-c5-0-15 Degree $2$ Conductor $700$ Sign $0.894 - 0.447i$ Analytic cond. $112.268$ Root an. cond. $10.5956$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 2i·3-s − 49i·7-s + 239·9-s − 720·11-s + 572i·13-s − 1.25e3i·17-s + 94·19-s − 98·21-s + 96i·23-s − 964i·27-s + 4.37e3·29-s − 6.24e3·31-s + 1.44e3i·33-s + 1.07e4i·37-s + 1.14e3·39-s + ⋯
 L(s)  = 1 − 0.128i·3-s − 0.377i·7-s + 0.983·9-s − 1.79·11-s + 0.938i·13-s − 1.05i·17-s + 0.0597·19-s − 0.0484·21-s + 0.0378i·23-s − 0.254i·27-s + 0.965·29-s − 1.16·31-s + 0.230i·33-s + 1.29i·37-s + 0.120·39-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$700$$    =    $$2^{2} \cdot 5^{2} \cdot 7$$ Sign: $0.894 - 0.447i$ Analytic conductor: $$112.268$$ Root analytic conductor: $$10.5956$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{700} (449, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 700,\ (\ :5/2),\ 0.894 - 0.447i)$$

Particular Values

 $$L(3)$$ $$\approx$$ $$1.751906671$$ $$L(\frac12)$$ $$\approx$$ $$1.751906671$$ $$L(\frac{7}{2})$$ 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$$
7 $$1 + 49iT$$
good3 $$1 + 2iT - 243T^{2}$$
11 $$1 + 720T + 1.61e5T^{2}$$
13 $$1 - 572iT - 3.71e5T^{2}$$
17 $$1 + 1.25e3iT - 1.41e6T^{2}$$
19 $$1 - 94T + 2.47e6T^{2}$$
23 $$1 - 96iT - 6.43e6T^{2}$$
29 $$1 - 4.37e3T + 2.05e7T^{2}$$
31 $$1 + 6.24e3T + 2.86e7T^{2}$$
37 $$1 - 1.07e4iT - 6.93e7T^{2}$$
41 $$1 - 1.20e4T + 1.15e8T^{2}$$
43 $$1 + 9.16e3iT - 1.47e8T^{2}$$
47 $$1 - 2.58e4iT - 2.29e8T^{2}$$
53 $$1 - 1.01e3iT - 4.18e8T^{2}$$
59 $$1 + 1.24e3T + 7.14e8T^{2}$$
61 $$1 - 7.59e3T + 8.44e8T^{2}$$
67 $$1 + 4.11e4iT - 1.35e9T^{2}$$
71 $$1 + 3.76e4T + 1.80e9T^{2}$$
73 $$1 + 1.34e4iT - 2.07e9T^{2}$$
79 $$1 + 6.24e3T + 3.07e9T^{2}$$
83 $$1 + 2.52e4iT - 3.93e9T^{2}$$
89 $$1 - 4.51e4T + 5.58e9T^{2}$$
97 $$1 + 1.07e5iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$