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Properties

Label 2-2736-19.18-c2-0-13
Degree $2$
Conductor $2736$
Sign $-0.595 - 0.803i$
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  + 3.78·5-s − 0.363·7-s − 8.03·11-s − 17.0i·13-s − 31.1·17-s + (11.3 + 15.2i)19-s + 34.4·23-s − 10.6·25-s + 5.21i·29-s − 4.14i·31-s − 1.37·35-s − 37.7i·37-s + 36.6i·41-s − 53.5·43-s + 23.4·47-s + ⋯
L(s)  = 1  + 0.757·5-s − 0.0519·7-s − 0.730·11-s − 1.31i·13-s − 1.83·17-s + (0.595 + 0.803i)19-s + 1.49·23-s − 0.426·25-s + 0.179i·29-s − 0.133i·31-s − 0.0393·35-s − 1.01i·37-s + 0.892i·41-s − 1.24·43-s + 0.499·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.595 - 0.803i$
Analytic conductor: \(74.5506\)
Root analytic conductor: \(8.63426\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1),\ -0.595 - 0.803i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7779012012\)
\(L(\frac12)\) \(\approx\) \(0.7779012012\)
\(L(2)\) 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 \( 1 + (-11.3 - 15.2i)T \)
good5 \( 1 - 3.78T + 25T^{2} \)
7 \( 1 + 0.363T + 49T^{2} \)
11 \( 1 + 8.03T + 121T^{2} \)
13 \( 1 + 17.0iT - 169T^{2} \)
17 \( 1 + 31.1T + 289T^{2} \)
23 \( 1 - 34.4T + 529T^{2} \)
29 \( 1 - 5.21iT - 841T^{2} \)
31 \( 1 + 4.14iT - 961T^{2} \)
37 \( 1 + 37.7iT - 1.36e3T^{2} \)
41 \( 1 - 36.6iT - 1.68e3T^{2} \)
43 \( 1 + 53.5T + 1.84e3T^{2} \)
47 \( 1 - 23.4T + 2.20e3T^{2} \)
53 \( 1 - 79.9iT - 2.80e3T^{2} \)
59 \( 1 - 31.2iT - 3.48e3T^{2} \)
61 \( 1 - 59.5T + 3.72e3T^{2} \)
67 \( 1 - 108. iT - 4.48e3T^{2} \)
71 \( 1 - 100. iT - 5.04e3T^{2} \)
73 \( 1 + 132.T + 5.32e3T^{2} \)
79 \( 1 - 14.6iT - 6.24e3T^{2} \)
83 \( 1 - 69.7T + 6.88e3T^{2} \)
89 \( 1 - 150. iT - 7.92e3T^{2} \)
97 \( 1 - 51.4iT - 9.40e3T^{2} \)
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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

−8.920582448166635958076207545074, −8.199853561123772909077808899983, −7.39747369397419663864007580448, −6.61621887564894051758800860445, −5.68917925698683188903206074064, −5.24496925597258187788612754086, −4.23081854221658191477807473112, −3.05529076840105584993375442503, −2.38141117508774768120049408665, −1.18523949401131762370136422223, 0.17222184394310597125892635449, 1.71654500114757261006807630036, 2.41403318256678000911221653189, 3.45699636336307119850876746126, 4.75143087774499168104676371131, 5.01495093702600478225103731490, 6.29449048548094121376571864724, 6.72653682686810389174985859968, 7.49411503793173777515890696903, 8.659624573246205832326867385280

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