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Properties

Label 2-2736-57.8-c1-0-0
Degree $2$
Conductor $2736$
Sign $-0.805 - 0.592i$
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  + (−0.524 − 0.302i)5-s − 4.37·7-s − 4.96i·11-s + (1.91 − 1.10i)13-s + (2.80 + 1.61i)17-s + (0.965 + 4.25i)19-s + (1.27 − 0.736i)23-s + (−2.31 − 4.01i)25-s + (1.05 + 1.82i)29-s − 9.85i·31-s + (2.29 + 1.32i)35-s + 5.03i·37-s + (−3.79 + 6.58i)41-s + (−1.50 + 2.60i)43-s + (−9.40 + 5.42i)47-s + ⋯
L(s)  = 1  + (−0.234 − 0.135i)5-s − 1.65·7-s − 1.49i·11-s + (0.530 − 0.306i)13-s + (0.679 + 0.392i)17-s + (0.221 + 0.975i)19-s + (0.266 − 0.153i)23-s + (−0.463 − 0.802i)25-s + (0.195 + 0.338i)29-s − 1.77i·31-s + (0.387 + 0.223i)35-s + 0.827i·37-s + (−0.593 + 1.02i)41-s + (−0.229 + 0.397i)43-s + (−1.37 + 0.791i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.805 - 0.592i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.805 - 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1036086235\)
\(L(\frac12)\) \(\approx\) \(0.1036086235\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.965 - 4.25i)T \)
good5 \( 1 + (0.524 + 0.302i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 4.37T + 7T^{2} \)
11 \( 1 + 4.96iT - 11T^{2} \)
13 \( 1 + (-1.91 + 1.10i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.80 - 1.61i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.27 + 0.736i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.05 - 1.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.85iT - 31T^{2} \)
37 \( 1 - 5.03iT - 37T^{2} \)
41 \( 1 + (3.79 - 6.58i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.50 - 2.60i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.40 - 5.42i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.08 + 1.87i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.306 - 0.530i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.45 + 7.72i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.7 - 6.77i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.90 - 3.29i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.75 - 3.04i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (13.7 + 7.96i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.1iT - 83T^{2} \)
89 \( 1 + (-6.78 - 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.12 - 1.80i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.167952999606253107153271550590, −8.214725795491008049983102382621, −7.895041899007291285356420409965, −6.52604010712148342221496259945, −6.17463771717130668369916174070, −5.52893657114971480112813274929, −4.16607416096646244504713194574, −3.37039409790127569177751826504, −2.91261588643189400650026421139, −1.17283196508469041552893198018, 0.03636280599995547123346067891, 1.66689680910376686402211255429, 2.95481371697090461042372057344, 3.52199434110470325790807347949, 4.54884520261691868437469804986, 5.43224504072295413868041634924, 6.38683145892346949974632189475, 7.13971172232564629943269137829, 7.37402447418810600562732505350, 8.793048057619456053284513170172

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