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Properties

Label 2-2736-76.27-c1-0-0
Degree $2$
Conductor $2736$
Sign $-0.980 - 0.195i$
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  + (−1.33 + 2.31i)5-s − 3.93i·7-s + 2.20i·11-s + (−3.60 + 2.07i)13-s + (0.571 − 0.990i)17-s + (4.24 − 0.990i)19-s + (3.19 − 1.84i)23-s + (−1.07 − 1.85i)25-s + (2.10 − 1.21i)29-s − 3.67·31-s + (9.11 + 5.25i)35-s + 10.0i·37-s + (−8.01 − 4.62i)41-s + (0.490 + 0.283i)43-s + (−10.6 + 6.13i)47-s + ⋯
L(s)  = 1  + (−0.597 + 1.03i)5-s − 1.48i·7-s + 0.664i·11-s + (−0.998 + 0.576i)13-s + (0.138 − 0.240i)17-s + (0.973 − 0.227i)19-s + (0.665 − 0.384i)23-s + (−0.214 − 0.371i)25-s + (0.390 − 0.225i)29-s − 0.659·31-s + (1.53 + 0.889i)35-s + 1.65i·37-s + (−1.25 − 0.723i)41-s + (0.0748 + 0.0432i)43-s + (−1.55 + 0.895i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.980 - 0.195i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.980 - 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3005713218\)
\(L(\frac12)\) \(\approx\) \(0.3005713218\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.24 + 0.990i)T \)
good5 \( 1 + (1.33 - 2.31i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.93iT - 7T^{2} \)
11 \( 1 - 2.20iT - 11T^{2} \)
13 \( 1 + (3.60 - 2.07i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.571 + 0.990i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.19 + 1.84i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.10 + 1.21i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.67T + 31T^{2} \)
37 \( 1 - 10.0iT - 37T^{2} \)
41 \( 1 + (8.01 + 4.62i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.490 - 0.283i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (10.6 - 6.13i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.09 - 2.36i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.33 - 2.31i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.41 + 11.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.73 + 3.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.24 - 10.8i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.35 + 2.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.50 + 6.07i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.98iT - 83T^{2} \)
89 \( 1 + (12.9 - 7.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.91 - 1.68i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.450163090568864874260244021084, −8.142221689131795020108684048758, −7.45020901376142636655907328290, −7.00630514434539308418809078396, −6.55582646225990263454472557540, −5.01890380643599263128611081978, −4.48811102213663056911219614905, −3.49408322054949813431025581017, −2.83513178742321873884247541638, −1.42510776327281485354783871707, 0.099159017778086131466454650459, 1.50344076811262636367668943632, 2.75601092221341191051832758769, 3.51801458054802008635993314799, 4.78885163752460325925538637372, 5.31679074602796782839383255804, 5.87949056149890481639491228891, 7.07128372312796260218743656810, 7.940713342822838077713115646497, 8.476228827676419962270409179822

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