# Properties

 Label 2-3072-48.5-c0-0-1 Degree $2$ Conductor $3072$ Sign $-0.382 - 0.923i$ Analytic cond. $1.53312$ Root an. cond. $1.23819$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.707 + 0.707i)3-s − 1.00i·9-s + (−1.41 + 1.41i)19-s + i·25-s + (0.707 + 0.707i)27-s + (1.41 + 1.41i)43-s + 49-s − 2.00i·57-s + (−1.41 + 1.41i)67-s + 2i·73-s + (−0.707 − 0.707i)75-s − 1.00·81-s − 2·97-s + ⋯
 L(s)  = 1 + (−0.707 + 0.707i)3-s − 1.00i·9-s + (−1.41 + 1.41i)19-s + i·25-s + (0.707 + 0.707i)27-s + (1.41 + 1.41i)43-s + 49-s − 2.00i·57-s + (−1.41 + 1.41i)67-s + 2i·73-s + (−0.707 − 0.707i)75-s − 1.00·81-s − 2·97-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3072$$    =    $$2^{10} \cdot 3$$ Sign: $-0.382 - 0.923i$ Analytic conductor: $$1.53312$$ Root analytic conductor: $$1.23819$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3072} (1793, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3072,\ (\ :0),\ -0.382 - 0.923i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.7304416220$$ $$L(\frac12)$$ $$\approx$$ $$0.7304416220$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (0.707 - 0.707i)T$$
good5 $$1 - iT^{2}$$
7 $$1 - T^{2}$$
11 $$1 - iT^{2}$$
13 $$1 - iT^{2}$$
17 $$1 - T^{2}$$
19 $$1 + (1.41 - 1.41i)T - iT^{2}$$
23 $$1 + T^{2}$$
29 $$1 + iT^{2}$$
31 $$1 + T^{2}$$
37 $$1 + iT^{2}$$
41 $$1 + T^{2}$$
43 $$1 + (-1.41 - 1.41i)T + iT^{2}$$
47 $$1 - T^{2}$$
53 $$1 - iT^{2}$$
59 $$1 - iT^{2}$$
61 $$1 - iT^{2}$$
67 $$1 + (1.41 - 1.41i)T - iT^{2}$$
71 $$1 + T^{2}$$
73 $$1 - 2iT - T^{2}$$
79 $$1 + T^{2}$$
83 $$1 + iT^{2}$$
89 $$1 + T^{2}$$
97 $$1 + 2T + 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

−9.188777729223045129511174371523, −8.501288829706252621221386772923, −7.61252048033858873517136605199, −6.71287564274634971063025448154, −5.94094145167146399279470691750, −5.43063973870499996027286541794, −4.32620683929509160746485781991, −3.89702041163973258740634789180, −2.74198755636911149032372621669, −1.36725538656361626900124952109, 0.51063105119599922290331032342, 1.95712338329204378325968463816, 2.72425410088202730598982268643, 4.16230907033899008918432432400, 4.81478377608992146015288375312, 5.75452046609404865511276035549, 6.42785414255528850372055491136, 7.05304989496546062757050854351, 7.79040635287870126992935266952, 8.624354187827936717032718184389