# Properties

 Degree 2 Conductor 11 Sign $0.844 + 0.534i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.02 − 1.41i)2-s + (2.68 − 8.25i)3-s + (4.00 + 12.3i)4-s + (−8.06 + 5.85i)5-s + (−8.89 − 12.2i)6-s + (−56.0 + 18.2i)7-s + (48.0 + 15.5i)8-s + (4.60 + 3.34i)9-s + 17.3i·10-s + (−16.1 − 119. i)11-s + 112.·12-s + (75.0 − 103. i)13-s + (−31.7 + 97.6i)14-s + (26.7 + 82.2i)15-s + (−96.6 + 70.1i)16-s + (22.8 + 31.4i)17-s + ⋯
 L(s)  = 1 + (0.256 − 0.352i)2-s + (0.297 − 0.917i)3-s + (0.250 + 0.770i)4-s + (−0.322 + 0.234i)5-s + (−0.246 − 0.339i)6-s + (−1.14 + 0.371i)7-s + (0.750 + 0.243i)8-s + (0.0568 + 0.0412i)9-s + 0.173i·10-s + (−0.133 − 0.991i)11-s + 0.781·12-s + (0.444 − 0.611i)13-s + (−0.161 + 0.498i)14-s + (0.118 + 0.365i)15-s + (−0.377 + 0.274i)16-s + (0.0790 + 0.108i)17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.534i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.844 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$11$$ $$\varepsilon$$ = $0.844 + 0.534i$ motivic weight = $$4$$ character : $\chi_{11} (7, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 11,\ (\ :2),\ 0.844 + 0.534i)$ $L(\frac{5}{2})$ $\approx$ $1.18135 - 0.342456i$ $L(\frac12)$ $\approx$ $1.18135 - 0.342456i$ $L(3)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 11$, $$F_p$$ is a polynomial of degree 2. If $p = 11$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad11 $$1 + (16.1 + 119. i)T$$
good2 $$1 + (-1.02 + 1.41i)T + (-4.94 - 15.2i)T^{2}$$
3 $$1 + (-2.68 + 8.25i)T + (-65.5 - 47.6i)T^{2}$$
5 $$1 + (8.06 - 5.85i)T + (193. - 594. i)T^{2}$$
7 $$1 + (56.0 - 18.2i)T + (1.94e3 - 1.41e3i)T^{2}$$
13 $$1 + (-75.0 + 103. i)T + (-8.82e3 - 2.71e4i)T^{2}$$
17 $$1 + (-22.8 - 31.4i)T + (-2.58e4 + 7.94e4i)T^{2}$$
19 $$1 + (560. + 182. i)T + (1.05e5 + 7.66e4i)T^{2}$$
23 $$1 - 973.T + 2.79e5T^{2}$$
29 $$1 + (141. - 46.0i)T + (5.72e5 - 4.15e5i)T^{2}$$
31 $$1 + (-140. - 102. i)T + (2.85e5 + 8.78e5i)T^{2}$$
37 $$1 + (-442. - 1.36e3i)T + (-1.51e6 + 1.10e6i)T^{2}$$
41 $$1 + (561. + 182. i)T + (2.28e6 + 1.66e6i)T^{2}$$
43 $$1 + 1.17e3iT - 3.41e6T^{2}$$
47 $$1 + (151. - 465. i)T + (-3.94e6 - 2.86e6i)T^{2}$$
53 $$1 + (1.80e3 + 1.31e3i)T + (2.43e6 + 7.50e6i)T^{2}$$
59 $$1 + (1.48e3 + 4.58e3i)T + (-9.80e6 + 7.12e6i)T^{2}$$
61 $$1 + (-2.83e3 - 3.90e3i)T + (-4.27e6 + 1.31e7i)T^{2}$$
67 $$1 - 187.T + 2.01e7T^{2}$$
71 $$1 + (537. - 390. i)T + (7.85e6 - 2.41e7i)T^{2}$$
73 $$1 + (-5.02e3 + 1.63e3i)T + (2.29e7 - 1.66e7i)T^{2}$$
79 $$1 + (5.01e3 - 6.90e3i)T + (-1.20e7 - 3.70e7i)T^{2}$$
83 $$1 + (136. + 187. i)T + (-1.46e7 + 4.51e7i)T^{2}$$
89 $$1 + 6.64e3T + 6.27e7T^{2}$$
97 $$1 + (5.91e3 + 4.29e3i)T + (2.73e7 + 8.41e7i)T^{2}$$
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\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}

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

−19.45913143690613832663684433886, −18.83542575268301966417944163363, −16.96916956745881516936708924623, −15.55513645699437981220831877192, −13.29227715448769113774337395939, −12.76176450493755462113213728412, −11.03277751230077122785601969880, −8.419517639833572874400981910865, −6.79519065506185683550935256931, −3.06720870789742373251382605435, 4.34097181921691043144127527094, 6.70588737493782200969618209835, 9.416578457041041045029976374748, 10.56565454183933163404492163689, 12.90122949381567115469815998797, 14.69419195063749924195016914148, 15.65094696498285305620470359071, 16.62756924985976010394693466098, 18.95793137659585725997738433466, 20.02621226478315407667460885660