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Properties

Label 2-2736-19.11-c1-0-37
Degree $2$
Conductor $2736$
Sign $-0.658 + 0.752i$
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  − 7-s − 2·11-s + (1.5 + 2.59i)13-s + (2 − 3.46i)17-s + (−4 − 1.73i)19-s + (−2 − 3.46i)23-s + (2.5 + 4.33i)25-s + 3·31-s − 5·37-s + (2 − 3.46i)41-s + (−4.5 + 7.79i)43-s + (−5 − 8.66i)47-s − 6·49-s + (−2 − 3.46i)53-s + (7 − 12.1i)59-s + ⋯
L(s)  = 1  − 0.377·7-s − 0.603·11-s + (0.416 + 0.720i)13-s + (0.485 − 0.840i)17-s + (−0.917 − 0.397i)19-s + (−0.417 − 0.722i)23-s + (0.5 + 0.866i)25-s + 0.538·31-s − 0.821·37-s + (0.312 − 0.541i)41-s + (−0.686 + 1.18i)43-s + (−0.729 − 1.26i)47-s − 0.857·49-s + (−0.274 − 0.475i)53-s + (0.911 − 1.57i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.658 + 0.752i$
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: \((2,\ 2736,\ (\ :1/2),\ -0.658 + 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6471937214\)
\(L(\frac12)\) \(\approx\) \(0.6471937214\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (-1.5 - 2.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 + (-2 + 3.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.5 - 7.79i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5 + 8.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7 + 12.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7 - 12.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.518480271794223725325165149378, −7.896486269005897439567859813887, −6.80695692885730810265991100203, −6.50234408314565223826133130662, −5.33513026917926569138637212403, −4.70758701232997568595353666594, −3.66021506325547834898682859232, −2.80731273971856373700670131271, −1.74217892660147439893482688745, −0.20797960963389775676988670256, 1.33785728387919940032344961685, 2.57520345403634591764875152523, 3.45691490806085514991615931488, 4.30514794206535342587555458358, 5.34283757442321885089513625322, 6.04492871575234126600456227681, 6.70053956205680381026279992250, 7.81917103414825971954623642986, 8.213496544847131166094314044002, 9.011708647492945420120669559810

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