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Properties

Label 2-2736-19.7-c1-0-40
Degree $2$
Conductor $2736$
Sign $-0.856 + 0.516i$
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  + (−1.65 − 2.85i)5-s − 1.44·7-s + 1.81·11-s + (0.5 − 0.866i)13-s + (3.30 + 5.71i)17-s + (−1 − 4.24i)19-s + (2.39 − 4.14i)23-s + (−2.94 + 5.10i)25-s + (4.78 − 8.28i)29-s + 4.55·31-s + (2.39 + 4.14i)35-s − 5.89·37-s + (1.48 + 2.57i)41-s + (−4.17 − 7.22i)43-s + (1.48 − 2.57i)47-s + ⋯
L(s)  = 1  + (−0.738 − 1.27i)5-s − 0.547·7-s + 0.547·11-s + (0.138 − 0.240i)13-s + (0.800 + 1.38i)17-s + (−0.229 − 0.973i)19-s + (0.498 − 0.864i)23-s + (−0.589 + 1.02i)25-s + (0.888 − 1.53i)29-s + 0.817·31-s + (0.404 + 0.700i)35-s − 0.969·37-s + (0.231 + 0.401i)41-s + (−0.636 − 1.10i)43-s + (0.216 − 0.374i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.856 + 0.516i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.856 + 0.516i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9816263894\)
\(L(\frac12)\) \(\approx\) \(0.9816263894\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 4.24i)T \)
good5 \( 1 + (1.65 + 2.85i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 1.44T + 7T^{2} \)
11 \( 1 - 1.81T + 11T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.30 - 5.71i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.39 + 4.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.78 + 8.28i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.55T + 31T^{2} \)
37 \( 1 + 5.89T + 37T^{2} \)
41 \( 1 + (-1.48 - 2.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.17 + 7.22i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.48 + 2.57i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.65 + 2.85i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.21 + 7.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.17 - 12.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.78 - 8.28i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.5 + 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.17 + 12.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.63T + 83T^{2} \)
89 \( 1 + (8.25 - 14.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.44 + 11.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.431854882932722705739375320267, −8.080173568945960708362840261607, −6.93500104435601145130740430796, −6.24742106321867829497895006251, −5.32537346021819970872976274628, −4.43983849594356380067889313474, −3.89476230239202835254394120667, −2.82275564538882994472971357899, −1.37159603277123062616988382934, −0.35244167096086664316184137844, 1.37474455466796349619980555794, 3.07076986235672709601776021540, 3.16544232375494819346943649335, 4.25545554546828768554245358538, 5.29858735637684629927578550219, 6.32583406868920099758094559385, 6.90449559201036933871420981140, 7.46122664879619792107389713502, 8.261406388578560256313244329751, 9.252211836966521854377740079000

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