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Properties

Label 2-2736-228.11-c1-0-15
Degree $2$
Conductor $2736$
Sign $0.206 - 0.978i$
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  + (3.43 + 1.98i)5-s + 2.01i·7-s + 3.45·11-s + (0.581 + 1.00i)13-s + (−1.94 − 1.12i)17-s + (−4.27 + 0.843i)19-s + (1.36 + 2.36i)23-s + (5.36 + 9.28i)25-s + (−0.0525 + 0.0303i)29-s + 2.92i·31-s + (−3.99 + 6.91i)35-s + 10.5·37-s + (9.71 + 5.60i)41-s + (−6.21 − 3.58i)43-s + (−5.99 − 10.3i)47-s + ⋯
L(s)  = 1  + (1.53 + 0.886i)5-s + 0.761i·7-s + 1.04·11-s + (0.161 + 0.279i)13-s + (−0.470 − 0.271i)17-s + (−0.981 + 0.193i)19-s + (0.284 + 0.493i)23-s + (1.07 + 1.85i)25-s + (−0.00976 + 0.00563i)29-s + 0.525i·31-s + (−0.674 + 1.16i)35-s + 1.73·37-s + (1.51 + 0.875i)41-s + (−0.947 − 0.547i)43-s + (−0.873 − 1.51i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.206 - 0.978i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.206 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.558285333\)
\(L(\frac12)\) \(\approx\) \(2.558285333\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.27 - 0.843i)T \)
good5 \( 1 + (-3.43 - 1.98i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 2.01iT - 7T^{2} \)
11 \( 1 - 3.45T + 11T^{2} \)
13 \( 1 + (-0.581 - 1.00i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.94 + 1.12i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.36 - 2.36i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.0525 - 0.0303i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.92iT - 31T^{2} \)
37 \( 1 - 10.5T + 37T^{2} \)
41 \( 1 + (-9.71 - 5.60i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.21 + 3.58i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.99 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.31 - 1.33i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.08 - 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.76 + 6.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.77 + 1.60i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.20 + 7.29i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.71 + 2.97i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.16 + 2.98i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + (12.2 - 7.04i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.03 + 8.72i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.222221037615555463255375953922, −8.436535582140115165639219058845, −7.25339577464247346765014822138, −6.40763415865161129554294389989, −6.18486898482572911249082390461, −5.30115954792335065212219958270, −4.26228972697177390730794593161, −3.09657229658660977525027501168, −2.29034823299742484970734407983, −1.51514853337270944254111187661, 0.848687649445795756366188552195, 1.71775926941931900800071502174, 2.72383449512836957844280398880, 4.20205883740556846420151259907, 4.56342612202851905007824024129, 5.76136975585478342072275388741, 6.24106365052370494735970540229, 6.94412891580328652975896192924, 8.097014023419976908145989512335, 8.764358168177030858837850477619

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