# Properties

 Label 2-3072-96.5-c0-0-5 Degree $2$ Conductor $3072$ Sign $0.195 - 0.980i$ 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 + (0.382 + 0.923i)3-s + (0.541 + 0.541i)7-s + (−0.707 + 0.707i)9-s + (1.70 − 0.707i)13-s + (−1.30 + 0.541i)19-s + (−0.292 + 0.707i)21-s + (0.707 + 0.707i)25-s + (−0.923 − 0.382i)27-s + 1.84·31-s + (−0.707 − 0.292i)37-s + (1.30 + 1.30i)39-s + (−0.541 + 1.30i)43-s − 0.414i·49-s + (−1 − 0.999i)57-s + (−0.707 − 1.70i)61-s + ⋯
 L(s)  = 1 + (0.382 + 0.923i)3-s + (0.541 + 0.541i)7-s + (−0.707 + 0.707i)9-s + (1.70 − 0.707i)13-s + (−1.30 + 0.541i)19-s + (−0.292 + 0.707i)21-s + (0.707 + 0.707i)25-s + (−0.923 − 0.382i)27-s + 1.84·31-s + (−0.707 − 0.292i)37-s + (1.30 + 1.30i)39-s + (−0.541 + 1.30i)43-s − 0.414i·49-s + (−1 − 0.999i)57-s + (−0.707 − 1.70i)61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3072$$    =    $$2^{10} \cdot 3$$ Sign: $0.195 - 0.980i$ Analytic conductor: $$1.53312$$ Root analytic conductor: $$1.23819$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3072} (2945, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3072,\ (\ :0),\ 0.195 - 0.980i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.546384809$$ $$L(\frac12)$$ $$\approx$$ $$1.546384809$$ $$L(1)$$ 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 + (-0.382 - 0.923i)T$$
good5 $$1 + (-0.707 - 0.707i)T^{2}$$
7 $$1 + (-0.541 - 0.541i)T + iT^{2}$$
11 $$1 + (0.707 + 0.707i)T^{2}$$
13 $$1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2}$$
17 $$1 + T^{2}$$
19 $$1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2}$$
23 $$1 + iT^{2}$$
29 $$1 + (0.707 - 0.707i)T^{2}$$
31 $$1 - 1.84T + T^{2}$$
37 $$1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2}$$
41 $$1 + iT^{2}$$
43 $$1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2}$$
47 $$1 + T^{2}$$
53 $$1 + (0.707 + 0.707i)T^{2}$$
59 $$1 + (-0.707 - 0.707i)T^{2}$$
61 $$1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2}$$
67 $$1 + (-0.707 + 0.707i)T^{2}$$
71 $$1 - iT^{2}$$
73 $$1 + (1 - i)T - iT^{2}$$
79 $$1 - 1.84iT - T^{2}$$
83 $$1 + (-0.707 + 0.707i)T^{2}$$
89 $$1 - iT^{2}$$
97 $$1 - 1.41T + 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

−8.877577586562685156155730356060, −8.331145441721883964650958857843, −8.053724219209599640982038412125, −6.62721595047507918918665091178, −5.91327057774597595842023209633, −5.15837647806033425390362341002, −4.34745715312598608404957164562, −3.52984534232785803063335439413, −2.72379420524008302338984415808, −1.52827055133606254980378963983, 1.02538880827801870979427363888, 1.92764796764071932154030066015, 2.98714584168916666603824810032, 4.00590364479828792775518949498, 4.70868297867080282857610122283, 6.04711232044242467508324236782, 6.49218553981150172575729028438, 7.17388694497541660879402764629, 8.109539461054794789270015396620, 8.635254787922849039787972318485