# Properties

 Label 2-11e2-11.4-c3-0-22 Degree $2$ Conductor $121$ Sign $-0.927 - 0.374i$ Analytic cond. $7.13923$ Root an. cond. $2.67193$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (2.47 − 7.60i)3-s + (−2.47 − 7.60i)4-s + (−14.5 + 10.5i)5-s + (−29.9 − 21.7i)9-s − 63.9·12-s + (44.4 + 136. i)15-s + (−51.7 + 37.6i)16-s + (116. + 84.6i)20-s − 108·23-s + (61.4 − 189. i)25-s + (−64.7 + 47.0i)27-s + (−275. − 199. i)31-s + (−91.4 + 281. i)36-s + (−134. − 412. i)37-s + 665.·45-s + ⋯
 L(s)  = 1 + (0.475 − 1.46i)3-s + (−0.309 − 0.951i)4-s + (−1.30 + 0.946i)5-s + (−1.10 − 0.805i)9-s − 1.53·12-s + (0.765 + 2.35i)15-s + (−0.809 + 0.587i)16-s + (1.30 + 0.946i)20-s − 0.979·23-s + (0.491 − 1.51i)25-s + (−0.461 + 0.335i)27-s + (−1.59 − 1.15i)31-s + (−0.423 + 1.30i)36-s + (−0.595 − 1.83i)37-s + 2.20·45-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$121$$    =    $$11^{2}$$ Sign: $-0.927 - 0.374i$ Analytic conductor: $$7.13923$$ Root analytic conductor: $$2.67193$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{121} (81, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 121,\ (\ :3/2),\ -0.927 - 0.374i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.122359 + 0.629957i$$ $$L(\frac12)$$ $$\approx$$ $$0.122359 + 0.629957i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad11 $$1$$
good2 $$1 + (2.47 + 7.60i)T^{2}$$
3 $$1 + (-2.47 + 7.60i)T + (-21.8 - 15.8i)T^{2}$$
5 $$1 + (14.5 - 10.5i)T + (38.6 - 118. i)T^{2}$$
7 $$1 + (-277. + 201. i)T^{2}$$
13 $$1 + (678. + 2.08e3i)T^{2}$$
17 $$1 + (1.51e3 - 4.67e3i)T^{2}$$
19 $$1 + (-5.54e3 - 4.03e3i)T^{2}$$
23 $$1 + 108T + 1.21e4T^{2}$$
29 $$1 + (-1.97e4 + 1.43e4i)T^{2}$$
31 $$1 + (275. + 199. i)T + (9.20e3 + 2.83e4i)T^{2}$$
37 $$1 + (134. + 412. i)T + (-4.09e4 + 2.97e4i)T^{2}$$
41 $$1 + (-5.57e4 - 4.05e4i)T^{2}$$
43 $$1 + 7.95e4T^{2}$$
47 $$1 + (11.1 - 34.2i)T + (-8.39e4 - 6.10e4i)T^{2}$$
53 $$1 + (-597. - 433. i)T + (4.60e4 + 1.41e5i)T^{2}$$
59 $$1 + (222. + 684. i)T + (-1.66e5 + 1.20e5i)T^{2}$$
61 $$1 + (7.01e4 - 2.15e5i)T^{2}$$
67 $$1 + 416T + 3.00e5T^{2}$$
71 $$1 + (495. - 359. i)T + (1.10e5 - 3.40e5i)T^{2}$$
73 $$1 + (-3.14e5 + 2.28e5i)T^{2}$$
79 $$1 + (1.52e5 + 4.68e5i)T^{2}$$
83 $$1 + (1.76e5 - 5.43e5i)T^{2}$$
89 $$1 - 1.67e3T + 7.04e5T^{2}$$
97 $$1 + (-27.5 - 19.9i)T + (2.82e5 + 8.68e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$