# Properties

 Label 2-42e2-12.11-c1-0-27 Degree $2$ Conductor $1764$ Sign $0.853 + 0.521i$ Analytic cond. $14.0856$ Root an. cond. $3.75308$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.03 − 0.965i)2-s + (0.133 + 1.99i)4-s − 2.73i·5-s + (1.78 − 2.19i)8-s + (−2.63 + 2.82i)10-s + 5.64·11-s + 1.41·13-s + (−3.96 + 0.534i)16-s + 6.19i·17-s + 4.13i·19-s + (5.45 − 0.366i)20-s + (−5.83 − 5.45i)22-s + 5.64·23-s − 2.46·25-s + (−1.46 − 1.36i)26-s + ⋯
 L(s)  = 1 + (−0.730 − 0.683i)2-s + (0.0669 + 0.997i)4-s − 1.22i·5-s + (0.632 − 0.774i)8-s + (−0.834 + 0.892i)10-s + 1.70·11-s + 0.392·13-s + (−0.991 + 0.133i)16-s + 1.50i·17-s + 0.947i·19-s + (1.21 − 0.0818i)20-s + (−1.24 − 1.16i)22-s + 1.17·23-s − 0.492·25-s + (−0.286 − 0.267i)26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.521i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1764$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{2}$$ Sign: $0.853 + 0.521i$ Analytic conductor: $$14.0856$$ Root analytic conductor: $$3.75308$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1764} (1079, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1764,\ (\ :1/2),\ 0.853 + 0.521i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.345385261$$ $$L(\frac12)$$ $$\approx$$ $$1.345385261$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.03 + 0.965i)T$$
3 $$1$$
7 $$1$$
good5 $$1 + 2.73iT - 5T^{2}$$
11 $$1 - 5.64T + 11T^{2}$$
13 $$1 - 1.41T + 13T^{2}$$
17 $$1 - 6.19iT - 17T^{2}$$
19 $$1 - 4.13iT - 19T^{2}$$
23 $$1 - 5.64T + 23T^{2}$$
29 $$1 - 0.378iT - 29T^{2}$$
31 $$1 - 7.15iT - 31T^{2}$$
37 $$1 + 3.46T + 37T^{2}$$
41 $$1 - 5.26iT - 41T^{2}$$
43 $$1 - 5.84iT - 43T^{2}$$
47 $$1 + 2.13T + 47T^{2}$$
53 $$1 + 0.656iT - 53T^{2}$$
59 $$1 + 13.8T + 59T^{2}$$
61 $$1 - 15.0T + 61T^{2}$$
67 $$1 + 7.98iT - 67T^{2}$$
71 $$1 - 13.9T + 71T^{2}$$
73 $$1 + 13.0T + 73T^{2}$$
79 $$1 - 13.8iT - 79T^{2}$$
83 $$1 - 10.1T + 83T^{2}$$
89 $$1 + 5.26iT - 89T^{2}$$
97 $$1 + 9.41T + 97T^{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

−9.167830690157723946231317095781, −8.567161747446572168667159033366, −8.129862682349496724452357884707, −6.90033604739247849695066332840, −6.17606353526347572174734219284, −4.90563235222907669637809625537, −4.02775666707237829878608608505, −3.34680426187155063433943586841, −1.59528905667869107886989077524, −1.19239835061893746890998074534, 0.804631926316138298542881284945, 2.24958134113888676503354150447, 3.34715352987697803388620731280, 4.53069638292010809357888116348, 5.57877002814039807622634211878, 6.59851644814309182136457535503, 6.90539005406248720782903913573, 7.53237436723783915127170820761, 8.790293634045720795055762714128, 9.216262946419771398179803493988