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Properties

Label 2-2736-228.83-c1-0-2
Degree $2$
Conductor $2736$
Sign $-0.752 + 0.658i$
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  + (−2.27 + 1.31i)5-s + 3.01i·7-s + 0.730·11-s + (2.20 − 3.81i)13-s + (−5.35 + 3.08i)17-s + (−2.91 + 3.24i)19-s + (−2.23 + 3.86i)23-s + (0.957 − 1.65i)25-s + (4.85 + 2.80i)29-s − 6.37i·31-s + (−3.96 − 6.87i)35-s − 1.48·37-s + (−3.54 + 2.04i)41-s + (6.82 − 3.93i)43-s + (1.28 − 2.22i)47-s + ⋯
L(s)  = 1  + (−1.01 + 0.587i)5-s + 1.14i·7-s + 0.220·11-s + (0.610 − 1.05i)13-s + (−1.29 + 0.749i)17-s + (−0.667 + 0.744i)19-s + (−0.465 + 0.806i)23-s + (0.191 − 0.331i)25-s + (0.900 + 0.520i)29-s − 1.14i·31-s + (−0.670 − 1.16i)35-s − 0.244·37-s + (−0.554 + 0.319i)41-s + (1.04 − 0.600i)43-s + (0.187 − 0.324i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.752 + 0.658i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.752 + 0.658i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1388374852\)
\(L(\frac12)\) \(\approx\) \(0.1388374852\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.91 - 3.24i)T \)
good5 \( 1 + (2.27 - 1.31i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.01iT - 7T^{2} \)
11 \( 1 - 0.730T + 11T^{2} \)
13 \( 1 + (-2.20 + 3.81i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.35 - 3.08i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.23 - 3.86i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.85 - 2.80i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.37iT - 31T^{2} \)
37 \( 1 + 1.48T + 37T^{2} \)
41 \( 1 + (3.54 - 2.04i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.82 + 3.93i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.28 + 2.22i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.22 + 5.32i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.08 - 1.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.26 - 7.39i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.40 - 4.85i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.00 + 8.67i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.72 + 9.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.78 - 1.02i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.34T + 83T^{2} \)
89 \( 1 + (2.22 + 1.28i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.80 + 4.86i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.069710111320764541194586270793, −8.425817730674650402251519646832, −7.955790711847139231404173224872, −7.03566875191228493711036017204, −6.13058230755458175286302559103, −5.65809711321232551167540839189, −4.41250830600874899221454456502, −3.68181169283173989582516494449, −2.84807521334146412594869478272, −1.78655135869434156097111260146, 0.04901034313653979410713754919, 1.14943783806405169025288049044, 2.52890277609036675829165924132, 3.83721273172365275354029482107, 4.38131260511601721366910669559, 4.79930812325568688524846325757, 6.40209053168248029975368351191, 6.79334847917196827586795401740, 7.59290607034616054526962181433, 8.491625222293328704604921123032

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