# Properties

 Label 2-1216-19.18-c2-0-77 Degree $2$ Conductor $1216$ Sign $1$ Analytic cond. $33.1336$ Root an. cond. $5.75617$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 5.65i·3-s − 7·5-s − 11·7-s − 23.0·9-s + 3·11-s − 11.3i·13-s + 39.5i·15-s − 17·17-s − 19·19-s + 62.2i·21-s − 2·23-s + 24·25-s + 79.1i·27-s − 39.5i·29-s + 5.65i·31-s + ⋯
 L(s)  = 1 − 1.88i·3-s − 1.40·5-s − 1.57·7-s − 2.55·9-s + 0.272·11-s − 0.870i·13-s + 2.63i·15-s − 17-s − 19-s + 2.96i·21-s − 0.0869·23-s + 0.959·25-s + 2.93i·27-s − 1.36i·29-s + 0.182i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1216$$    =    $$2^{6} \cdot 19$$ Sign: $1$ Analytic conductor: $$33.1336$$ Root analytic conductor: $$5.75617$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{1216} (1025, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1216,\ (\ :1),\ 1)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.06792195590$$ $$L(\frac12)$$ $$\approx$$ $$0.06792195590$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
19 $$1 + 19T$$
good3 $$1 + 5.65iT - 9T^{2}$$
5 $$1 + 7T + 25T^{2}$$
7 $$1 + 11T + 49T^{2}$$
11 $$1 - 3T + 121T^{2}$$
13 $$1 + 11.3iT - 169T^{2}$$
17 $$1 + 17T + 289T^{2}$$
23 $$1 + 2T + 529T^{2}$$
29 $$1 + 39.5iT - 841T^{2}$$
31 $$1 - 5.65iT - 961T^{2}$$
37 $$1 + 39.5iT - 1.36e3T^{2}$$
41 $$1 + 39.5iT - 1.68e3T^{2}$$
43 $$1 + 21T + 1.84e3T^{2}$$
47 $$1 - 5T + 2.20e3T^{2}$$
53 $$1 - 5.65iT - 2.80e3T^{2}$$
59 $$1 + 33.9iT - 3.48e3T^{2}$$
61 $$1 + 23T + 3.72e3T^{2}$$
67 $$1 + 39.5iT - 4.48e3T^{2}$$
71 $$1 - 90.5iT - 5.04e3T^{2}$$
73 $$1 - 39T + 5.32e3T^{2}$$
79 $$1 + 96.1iT - 6.24e3T^{2}$$
83 $$1 + 6T + 6.88e3T^{2}$$
89 $$1 + 118. iT - 7.92e3T^{2}$$
97 $$1 - 169. iT - 9.40e3T^{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

−8.431825644146934926656598884659, −7.80743403561272948359982110156, −7.02308606120827680893592290552, −6.50506623033683166689465842016, −5.74116235583849464221598370865, −4.08562084291334047450648913169, −3.15777091138946962357321195091, −2.21794971763085243798823570837, −0.48773837777327157574429027387, −0.04288121501771759171089913368, 2.82158798408060234573502600703, 3.64239698037817293357079682601, 4.16865018399797391887960700234, 4.89053807554149028884187376236, 6.24383336522533258810506020173, 6.87427926301022186856762226721, 8.277787125946209191810643769546, 8.938076511698764801202415780583, 9.499504841946765614837880419047, 10.31150705842521774838538054939