Properties

Label 2-2736-4.3-c2-0-42
Degree $2$
Conductor $2736$
Sign $0.866 - 0.5i$
Analytic cond. $74.5506$
Root an. cond. $8.63426$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.54·5-s + 0.687i·7-s − 12.1i·11-s − 8.29·13-s + 13.7·17-s + 4.35i·19-s + 42.5i·23-s + 17.8·25-s − 53.4·29-s + 36.9i·31-s + 4.49i·35-s + 24.7·37-s + 79.9·41-s − 4.56i·43-s + 78.0i·47-s + ⋯
L(s)  = 1  + 1.30·5-s + 0.0981i·7-s − 1.10i·11-s − 0.638·13-s + 0.811·17-s + 0.229i·19-s + 1.84i·23-s + 0.713·25-s − 1.84·29-s + 1.19i·31-s + 0.128i·35-s + 0.668·37-s + 1.95·41-s − 0.106i·43-s + 1.65i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.866 - 0.5i$
Analytic conductor: \(74.5506\)
Root analytic conductor: \(8.63426\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1711, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1),\ 0.866 - 0.5i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.674146995\)
\(L(\frac12)\) \(\approx\) \(2.674146995\)
\(L(2)\) 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 \( 1 - 4.35iT \)
good5 \( 1 - 6.54T + 25T^{2} \)
7 \( 1 - 0.687iT - 49T^{2} \)
11 \( 1 + 12.1iT - 121T^{2} \)
13 \( 1 + 8.29T + 169T^{2} \)
17 \( 1 - 13.7T + 289T^{2} \)
23 \( 1 - 42.5iT - 529T^{2} \)
29 \( 1 + 53.4T + 841T^{2} \)
31 \( 1 - 36.9iT - 961T^{2} \)
37 \( 1 - 24.7T + 1.36e3T^{2} \)
41 \( 1 - 79.9T + 1.68e3T^{2} \)
43 \( 1 + 4.56iT - 1.84e3T^{2} \)
47 \( 1 - 78.0iT - 2.20e3T^{2} \)
53 \( 1 - 13.8T + 2.80e3T^{2} \)
59 \( 1 - 55.3iT - 3.48e3T^{2} \)
61 \( 1 - 43.2T + 3.72e3T^{2} \)
67 \( 1 + 112. iT - 4.48e3T^{2} \)
71 \( 1 + 8.14iT - 5.04e3T^{2} \)
73 \( 1 - 35.2T + 5.32e3T^{2} \)
79 \( 1 - 4.57iT - 6.24e3T^{2} \)
83 \( 1 - 50.9iT - 6.88e3T^{2} \)
89 \( 1 - 107.T + 7.92e3T^{2} \)
97 \( 1 - 98.4T + 9.40e3T^{2} \)
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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

−9.001236753627708365090452879945, −7.80933884013158378708183319209, −7.33823339263219447026363874466, −6.05160888879031162511961754893, −5.78061937532769382288454551528, −5.09661969373563648433871196696, −3.79035208207266138666215033645, −2.95398052046308497610658440256, −1.95390543034345822906236576246, −1.00410622157216999863226767527, 0.68753873819545242601733972737, 2.12535139297494721356185204158, 2.40167656755429728437271533171, 3.88962599727634422705503102511, 4.74286061938298262053309152623, 5.57404120584780904669223731665, 6.17481850339186109363531747321, 7.12751393815486837562993874991, 7.65777974379092012072912147567, 8.748214152657157370005359436900

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