Properties

 Label 2-2736-57.50-c1-0-5 Degree $2$ Conductor $2736$ Sign $-0.953 - 0.300i$ Analytic cond. $21.8470$ Root an. cond. $4.67408$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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Dirichlet series

 L(s)  = 1 + (−1.22 + 0.707i)5-s − 1.44·7-s + 0.635i·11-s + (5.17 + 2.98i)13-s + (1.77 − 1.02i)17-s + (−4 + 1.73i)19-s + (−4.89 − 2.82i)23-s + (−1.50 + 2.59i)25-s + (1.22 − 2.12i)29-s + 0.953i·31-s + (1.77 − 1.02i)35-s − 2.51i·37-s + (1.94 + 3.37i)43-s + (1.77 + 1.02i)47-s − 4.89·49-s + ⋯
 L(s)  = 1 + (−0.547 + 0.316i)5-s − 0.547·7-s + 0.191i·11-s + (1.43 + 0.828i)13-s + (0.430 − 0.248i)17-s + (−0.917 + 0.397i)19-s + (−1.02 − 0.589i)23-s + (−0.300 + 0.519i)25-s + (0.227 − 0.393i)29-s + 0.171i·31-s + (0.300 − 0.173i)35-s − 0.412i·37-s + (0.297 + 0.514i)43-s + (0.258 + 0.149i)47-s − 0.699·49-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$2736$$    =    $$2^{4} \cdot 3^{2} \cdot 19$$ Sign: $-0.953 - 0.300i$ Analytic conductor: $$21.8470$$ Root analytic conductor: $$4.67408$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{2736} (449, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2736,\ (\ :1/2),\ -0.953 - 0.300i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.4922472627$$ $$L(\frac12)$$ $$\approx$$ $$0.4922472627$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
19 $$1 + (4 - 1.73i)T$$
good5 $$1 + (1.22 - 0.707i)T + (2.5 - 4.33i)T^{2}$$
7 $$1 + 1.44T + 7T^{2}$$
11 $$1 - 0.635iT - 11T^{2}$$
13 $$1 + (-5.17 - 2.98i)T + (6.5 + 11.2i)T^{2}$$
17 $$1 + (-1.77 + 1.02i)T + (8.5 - 14.7i)T^{2}$$
23 $$1 + (4.89 + 2.82i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 + (-1.22 + 2.12i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 - 0.953iT - 31T^{2}$$
37 $$1 + 2.51iT - 37T^{2}$$
41 $$1 + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (-1.94 - 3.37i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (-1.77 - 1.02i)T + (23.5 + 40.7i)T^{2}$$
53 $$1 + (2.44 - 4.24i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (7.22 + 12.5i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-1.72 + 2.98i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-8.84 - 5.10i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2}$$
73 $$1 + (-1.05 - 1.81i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (0.825 - 0.476i)T + (39.5 - 68.4i)T^{2}$$
83 $$1 - 11.8iT - 83T^{2}$$
89 $$1 + (6.12 - 10.6i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (16.3 - 9.43i)T + (48.5 - 84.0i)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

−9.232486564416804662213812925329, −8.244816290386869379989530880281, −7.83887326168765573273495511475, −6.59825824802524399108564864640, −6.42154097953773391144489428136, −5.36880355583514362488468680838, −4.08562092367095796951506087054, −3.80260676877354525008802544650, −2.64571557677682323745443393841, −1.46362198568462119419152114756, 0.16438642888878608902395100059, 1.44449615105317904145381037266, 2.85211178104411688952897683970, 3.71469928791547836973468570659, 4.32512054519750709689735791869, 5.56030380873662557644912392553, 6.10155366383122081620147513479, 6.91509893632376570737720528935, 8.014939842778564576009406032316, 8.314629303462134389570418859092