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Properties

Label 2-2736-19.11-c1-0-1
Degree $2$
Conductor $2736$
Sign $-0.412 + 0.910i$
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  + (−1.95 + 3.39i)5-s − 2.04·7-s − 1.91·11-s + (2.13 + 3.68i)13-s + (−3.04 + 5.27i)17-s + (−4.33 − 0.497i)19-s + (3.42 + 5.93i)23-s + (−5.17 − 8.96i)25-s + (−1.34 − 2.32i)29-s + 1.46·31-s + (4.00 − 6.93i)35-s + 3.93·37-s + (−1.70 + 2.94i)41-s + (3.77 − 6.54i)43-s + (−4.96 − 8.59i)47-s + ⋯
L(s)  = 1  + (−0.875 + 1.51i)5-s − 0.772·7-s − 0.578·11-s + (0.590 + 1.02i)13-s + (−0.738 + 1.27i)17-s + (−0.993 − 0.114i)19-s + (0.714 + 1.23i)23-s + (−1.03 − 1.79i)25-s + (−0.249 − 0.431i)29-s + 0.263·31-s + (0.676 − 1.17i)35-s + 0.647·37-s + (−0.265 + 0.459i)41-s + (0.576 − 0.997i)43-s + (−0.723 − 1.25i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.412 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.412 + 0.910i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.412 + 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2453383276\)
\(L(\frac12)\) \(\approx\) \(0.2453383276\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.33 + 0.497i)T \)
good5 \( 1 + (1.95 - 3.39i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 2.04T + 7T^{2} \)
11 \( 1 + 1.91T + 11T^{2} \)
13 \( 1 + (-2.13 - 3.68i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.04 - 5.27i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.42 - 5.93i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.34 + 2.32i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.46T + 31T^{2} \)
37 \( 1 - 3.93T + 37T^{2} \)
41 \( 1 + (1.70 - 2.94i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.77 + 6.54i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.96 + 8.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.91 - 8.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.61 + 2.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.70 + 8.14i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.28 - 9.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.21 - 3.84i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.33 - 10.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.52 + 11.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.43T + 83T^{2} \)
89 \( 1 + (3.47 + 6.01i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.58 + 4.48i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.333156315511261821147044753000, −8.491783354674369156285634511081, −7.76803571886164522622522905123, −6.85367313693910163800913943595, −6.57464696662379560518451379386, −5.74030374954582782398843434879, −4.28181406214719657357245897008, −3.75875237536006266334224423493, −2.93629979804843107180098123639, −1.97252546452367511024232805248, 0.097911899873127361813723475257, 0.925219614894768588195028668963, 2.56314928537705779068526251770, 3.48437794505205816118409342454, 4.53402105482719175160808692458, 4.94461685629829631686939587823, 5.94745507698616906483110504658, 6.80532855191528329815353834771, 7.78303113719281520676540194234, 8.327039618600249055876065498945

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