Properties

Label 2-2736-57.8-c1-0-34
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 $0$

Origins

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Normalization:  

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: $\chi_{2736} (1889, \cdot )$
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

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

Graph of the $Z$-function along the critical line