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Properties

Label 2-2736-19.7-c1-0-9
Degree $2$
Conductor $2736$
Sign $-0.899 - 0.437i$
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  + (1.17 + 2.02i)5-s − 3.83·7-s + 3.34·11-s + (−3.08 + 5.34i)13-s + (2.59 + 4.50i)17-s + (3.01 − 3.14i)19-s + (−1.17 + 2.02i)23-s + (−0.244 + 0.423i)25-s + (0.0250 − 0.0434i)29-s + 3.43·31-s + (−4.48 − 7.77i)35-s − 5.43·37-s + (−3.64 − 6.31i)41-s + (−4.43 − 7.67i)43-s + (−5.36 + 9.29i)47-s + ⋯
L(s)  = 1  + (0.523 + 0.907i)5-s − 1.44·7-s + 1.00·11-s + (−0.856 + 1.48i)13-s + (0.630 + 1.09i)17-s + (0.691 − 0.722i)19-s + (−0.244 + 0.423i)23-s + (−0.0489 + 0.0847i)25-s + (0.00466 − 0.00807i)29-s + 0.617·31-s + (−0.758 − 1.31i)35-s − 0.894·37-s + (−0.569 − 0.986i)41-s + (−0.675 − 1.17i)43-s + (−0.783 + 1.35i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.899 - 0.437i$
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: \((2,\ 2736,\ (\ :1/2),\ -0.899 - 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.017088864\)
\(L(\frac12)\) \(\approx\) \(1.017088864\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.01 + 3.14i)T \)
good5 \( 1 + (-1.17 - 2.02i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.83T + 7T^{2} \)
11 \( 1 - 3.34T + 11T^{2} \)
13 \( 1 + (3.08 - 5.34i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.59 - 4.50i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.17 - 2.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.0250 + 0.0434i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.43T + 31T^{2} \)
37 \( 1 + 5.43T + 37T^{2} \)
41 \( 1 + (3.64 + 6.31i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.43 + 7.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.36 - 9.29i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.59 - 2.76i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.92 + 3.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.46 - 2.53i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.75 - 9.97i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.744 + 1.28i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.84 - 6.65i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.0875 - 0.151i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.00T + 83T^{2} \)
89 \( 1 + (6.77 - 11.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.84 - 3.19i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.391590415620753487136940832924, −8.572665688297385851286314467314, −7.27575868127544790524871371259, −6.75031966525609089897460557730, −6.37312319338804193165201994167, −5.46957391688592352903961302833, −4.22536350273921261510173141294, −3.46456621445416204803558983669, −2.65291303451834435554026003408, −1.57506651254605296634772252973, 0.32750099656648494969544384122, 1.43826425333825599818937882611, 2.95696627051738520714635018284, 3.41099635813918224858547667214, 4.75615735146348148548041409403, 5.38348131893205184473243155297, 6.16568094804870054525835945256, 6.89548195701132229736672936521, 7.77935932562736346734754442912, 8.588275469673017508743184541301

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