# Properties

 Label 2-11e2-11.9-c3-0-2 Degree $2$ Conductor $121$ Sign $-0.944 - 0.329i$ 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 + (−6.47 + 4.70i)3-s + (6.47 + 4.70i)4-s + (5.56 + 17.1i)5-s + (11.4 − 35.1i)9-s − 64·12-s + (−116. − 84.6i)15-s + (19.7 + 60.8i)16-s + (−44.4 + 136. i)20-s − 108·23-s + (−160. + 116. i)25-s + (24.7 + 76.0i)27-s + (105. − 323. i)31-s + (239. − 173. i)36-s + (351. + 255. i)37-s + 666·45-s + ⋯
 L(s)  = 1 + (−1.24 + 0.904i)3-s + (0.809 + 0.587i)4-s + (0.497 + 1.53i)5-s + (0.423 − 1.30i)9-s − 1.53·12-s + (−2.00 − 1.45i)15-s + (0.309 + 0.951i)16-s + (−0.497 + 1.53i)20-s − 0.979·23-s + (−1.28 + 0.935i)25-s + (0.176 + 0.542i)27-s + (0.608 − 1.87i)31-s + (1.10 − 0.805i)36-s + (1.56 + 1.13i)37-s + 2.20·45-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$121$$    =    $$11^{2}$$ Sign: $-0.944 - 0.329i$ Analytic conductor: $$7.13923$$ Root analytic conductor: $$2.67193$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{121} (9, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 121,\ (\ :3/2),\ -0.944 - 0.329i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.201844 + 1.18960i$$ $$L(\frac12)$$ $$\approx$$ $$0.201844 + 1.18960i$$ $$L(\frac{5}{2})$$ 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 + (-6.47 - 4.70i)T^{2}$$
3 $$1 + (6.47 - 4.70i)T + (8.34 - 25.6i)T^{2}$$
5 $$1 + (-5.56 - 17.1i)T + (-101. + 73.4i)T^{2}$$
7 $$1 + (105. + 326. i)T^{2}$$
13 $$1 + (-1.77e3 - 1.29e3i)T^{2}$$
17 $$1 + (-3.97e3 + 2.88e3i)T^{2}$$
19 $$1 + (2.11e3 - 6.52e3i)T^{2}$$
23 $$1 + 108T + 1.21e4T^{2}$$
29 $$1 + (7.53e3 + 2.31e4i)T^{2}$$
31 $$1 + (-105. + 323. i)T + (-2.41e4 - 1.75e4i)T^{2}$$
37 $$1 + (-351. - 255. i)T + (1.56e4 + 4.81e4i)T^{2}$$
41 $$1 + (2.12e4 - 6.55e4i)T^{2}$$
43 $$1 + 7.95e4T^{2}$$
47 $$1 + (-29.1 + 21.1i)T + (3.20e4 - 9.87e4i)T^{2}$$
53 $$1 + (228. - 701. i)T + (-1.20e5 - 8.75e4i)T^{2}$$
59 $$1 + (-582. - 423. i)T + (6.34e4 + 1.95e5i)T^{2}$$
61 $$1 + (-1.83e5 + 1.33e5i)T^{2}$$
67 $$1 + 416T + 3.00e5T^{2}$$
71 $$1 + (-189. - 582. i)T + (-2.89e5 + 2.10e5i)T^{2}$$
73 $$1 + (1.20e5 + 3.69e5i)T^{2}$$
79 $$1 + (-3.98e5 - 2.89e5i)T^{2}$$
83 $$1 + (-4.62e5 + 3.36e5i)T^{2}$$
89 $$1 - 1.67e3T + 7.04e5T^{2}$$
97 $$1 + (10.5 - 32.3i)T + (-7.38e5 - 5.36e5i)T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−13.39287271661242716528364267142, −11.86357081068329006659452105624, −11.33587029423411038573644578691, −10.48125300160710025761438972615, −9.803193952001627024248566536668, −7.74610184048241485852764267269, −6.49318848123488723063215350588, −5.90503619536109020927685983675, −4.06522403580169364017072692867, −2.59746914012761409600697914585, 0.73736457287548767861729506349, 1.80673807592092051780541294447, 4.92532901675372698611808546685, 5.77514625800853500204102325643, 6.63569402112610197700212531436, 7.985445085622730871184504535243, 9.472763262618668507949373388747, 10.65235589622745417810869971896, 11.73192697499522264578332450728, 12.39013762170305326752225823079