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Properties

Label 2-2736-19.7-c1-0-17
Degree $2$
Conductor $2736$
Sign $0.0977 - 0.995i$
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 + 1.73i)5-s − 3·7-s + 2·11-s + (0.5 − 0.866i)13-s + (3 + 5.19i)17-s + (4 + 1.73i)19-s + (2 − 3.46i)23-s + (0.500 − 0.866i)25-s + (−1 + 1.73i)29-s − 7·31-s + (−3 − 5.19i)35-s + 37-s + (−4 − 6.92i)41-s + (3.5 + 6.06i)43-s + (−4 + 6.92i)47-s + ⋯
L(s)  = 1  + (0.447 + 0.774i)5-s − 1.13·7-s + 0.603·11-s + (0.138 − 0.240i)13-s + (0.727 + 1.26i)17-s + (0.917 + 0.397i)19-s + (0.417 − 0.722i)23-s + (0.100 − 0.173i)25-s + (−0.185 + 0.321i)29-s − 1.25·31-s + (−0.507 − 0.878i)35-s + 0.164·37-s + (−0.624 − 1.08i)41-s + (0.533 + 0.924i)43-s + (−0.583 + 1.01i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.0977 - 0.995i$
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: \((2,\ 2736,\ (\ :1/2),\ 0.0977 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.637407216\)
\(L(\frac12)\) \(\approx\) \(1.637407216\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 3T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + (4 + 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.5 - 6.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4 + 6.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.5 - 7.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1 + 1.73i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.5 - 12.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7 - 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.125290788127427790816947691961, −8.278222918047085923496680445435, −7.31295227896372050019358837635, −6.65740399856390852467740553797, −6.04027037726084458935113279692, −5.37682671200619408122051014423, −3.95328821760891640198707889357, −3.37784190423396212618533413164, −2.50536182422587660507874318240, −1.20703424340585459349894803687, 0.58593917508299354324145172196, 1.71234140565390077599417743962, 3.07644033522704432918393632944, 3.66211487330737531111729737125, 4.95589979745193449120180326619, 5.43230550330623335016838517610, 6.37842240954551453391717228906, 7.09087660869865506366533887771, 7.80638534250552728890218063465, 9.048010214949952576064860281132

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