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Properties

Label 2-2736-76.31-c1-0-10
Degree $2$
Conductor $2736$
Sign $0.671 - 0.740i$
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  + 1.73i·7-s − 3.46i·11-s + (−4.5 − 2.59i)13-s + (3 + 5.19i)17-s + (−4 + 1.73i)19-s + (3 + 1.73i)23-s + (2.5 − 4.33i)25-s + (3 + 1.73i)29-s + 31-s + 8.66i·37-s + (6 − 3.46i)41-s + (−4.5 + 2.59i)43-s + (9 + 5.19i)47-s + 4·49-s + (9 + 5.19i)53-s + ⋯
L(s)  = 1  + 0.654i·7-s − 1.04i·11-s + (−1.24 − 0.720i)13-s + (0.727 + 1.26i)17-s + (−0.917 + 0.397i)19-s + (0.625 + 0.361i)23-s + (0.5 − 0.866i)25-s + (0.557 + 0.321i)29-s + 0.179·31-s + 1.42i·37-s + (0.937 − 0.541i)41-s + (−0.686 + 0.396i)43-s + (1.31 + 0.757i)47-s + 0.571·49-s + (1.23 + 0.713i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.671 - 0.740i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1855, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.671 - 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.550413757\)
\(L(\frac12)\) \(\approx\) \(1.550413757\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 1.73iT - 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + (4.5 + 2.59i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3 - 1.73i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 1.73i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 - 8.66iT - 37T^{2} \)
41 \( 1 + (-6 + 3.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.5 - 2.59i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9 - 5.19i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9 - 5.19i)T + (26.5 + 45.8i)T^{2} \)
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 \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 17.3iT - 83T^{2} \)
89 \( 1 + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12 + 6.92i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.687060378136006594473080065015, −8.327453372297320095684435356319, −7.53848099665972617328565541188, −6.49430988263262441817710785932, −5.84211032669462278804462405690, −5.16546840831288235276245419748, −4.17151558810655660165013039338, −3.11494196354921670215982180470, −2.41461399522982433159967505647, −0.998996254448405603439831032123, 0.60542505835964058463003449133, 2.05495924288036352418961036127, 2.86831235986725589289400852387, 4.16682532921498144871198831288, 4.69493644329183855329332637383, 5.48002251931227763329425085367, 6.75174274607511829942234768190, 7.21966101702330435056828786085, 7.66350171950873883741614778833, 9.014214153243778854867179585060

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