Properties

 Label 2-69-69.68-c5-0-9 Degree $2$ Conductor $69$ Sign $-0.192 - 0.981i$ Analytic cond. $11.0664$ Root an. cond. $3.32663$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 1.27i·2-s + (11.5 + 10.5i)3-s + 30.3·4-s − 80.0·5-s + (13.4 − 14.6i)6-s + 196. i·7-s − 79.5i·8-s + (21.7 + 242. i)9-s + 102. i·10-s + 20.8·11-s + (349. + 319. i)12-s − 791.·13-s + 251.·14-s + (−921. − 842. i)15-s + 870.·16-s + 174.·17-s + ⋯
 L(s)  = 1 − 0.225i·2-s + (0.738 + 0.674i)3-s + 0.949·4-s − 1.43·5-s + (0.152 − 0.166i)6-s + 1.51i·7-s − 0.439i·8-s + (0.0895 + 0.995i)9-s + 0.323i·10-s + 0.0520·11-s + (0.700 + 0.640i)12-s − 1.29·13-s + 0.342·14-s + (−1.05 − 0.966i)15-s + 0.849·16-s + 0.146·17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$69$$    =    $$3 \cdot 23$$ Sign: $-0.192 - 0.981i$ Analytic conductor: $$11.0664$$ Root analytic conductor: $$3.32663$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{69} (68, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 69,\ (\ :5/2),\ -0.192 - 0.981i)$$

Particular Values

 $$L(3)$$ $$\approx$$ $$1.16725 + 1.41784i$$ $$L(\frac12)$$ $$\approx$$ $$1.16725 + 1.41784i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-11.5 - 10.5i)T$$
23 $$1 + (-2.03e3 - 1.50e3i)T$$
good2 $$1 + 1.27iT - 32T^{2}$$
5 $$1 + 80.0T + 3.12e3T^{2}$$
7 $$1 - 196. iT - 1.68e4T^{2}$$
11 $$1 - 20.8T + 1.61e5T^{2}$$
13 $$1 + 791.T + 3.71e5T^{2}$$
17 $$1 - 174.T + 1.41e6T^{2}$$
19 $$1 - 892. iT - 2.47e6T^{2}$$
29 $$1 - 5.42e3iT - 2.05e7T^{2}$$
31 $$1 - 3.74e3T + 2.86e7T^{2}$$
37 $$1 + 1.07e4iT - 6.93e7T^{2}$$
41 $$1 - 5.10e3iT - 1.15e8T^{2}$$
43 $$1 + 1.01e4iT - 1.47e8T^{2}$$
47 $$1 - 2.20e3iT - 2.29e8T^{2}$$
53 $$1 - 3.84e4T + 4.18e8T^{2}$$
59 $$1 + 4.96e4iT - 7.14e8T^{2}$$
61 $$1 - 2.88e4iT - 8.44e8T^{2}$$
67 $$1 + 6.11e4iT - 1.35e9T^{2}$$
71 $$1 + 1.17e4iT - 1.80e9T^{2}$$
73 $$1 - 4.87e4T + 2.07e9T^{2}$$
79 $$1 - 6.58e4iT - 3.07e9T^{2}$$
83 $$1 - 5.35e4T + 3.93e9T^{2}$$
89 $$1 + 1.06e5T + 5.58e9T^{2}$$
97 $$1 - 5.96e4iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$