Properties

Label 2-2736-19.18-c2-0-61
Degree $2$
Conductor $2736$
Sign $0.923 - 0.382i$
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  + 8.20·5-s + 8.00·7-s + 7.74·11-s + 7.64i·13-s − 11.4·17-s + (−17.5 + 7.27i)19-s + 22.9·23-s + 42.3·25-s + 17.5i·29-s − 52.3i·31-s + 65.7·35-s + 40.7i·37-s + 36.2i·41-s + 52.5·43-s + 3.90·47-s + ⋯
L(s)  = 1  + 1.64·5-s + 1.14·7-s + 0.703·11-s + 0.588i·13-s − 0.675·17-s + (−0.923 + 0.382i)19-s + 0.998·23-s + 1.69·25-s + 0.606i·29-s − 1.68i·31-s + 1.87·35-s + 1.10i·37-s + 0.883i·41-s + 1.22·43-s + 0.0831·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.923 - 0.382i$
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: \((2,\ 2736,\ (\ :1),\ 0.923 - 0.382i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.745080713\)
\(L(\frac12)\) \(\approx\) \(3.745080713\)
\(L(2)\) 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 \( 1 + (17.5 - 7.27i)T \)
good5 \( 1 - 8.20T + 25T^{2} \)
7 \( 1 - 8.00T + 49T^{2} \)
11 \( 1 - 7.74T + 121T^{2} \)
13 \( 1 - 7.64iT - 169T^{2} \)
17 \( 1 + 11.4T + 289T^{2} \)
23 \( 1 - 22.9T + 529T^{2} \)
29 \( 1 - 17.5iT - 841T^{2} \)
31 \( 1 + 52.3iT - 961T^{2} \)
37 \( 1 - 40.7iT - 1.36e3T^{2} \)
41 \( 1 - 36.2iT - 1.68e3T^{2} \)
43 \( 1 - 52.5T + 1.84e3T^{2} \)
47 \( 1 - 3.90T + 2.20e3T^{2} \)
53 \( 1 - 49.8iT - 2.80e3T^{2} \)
59 \( 1 - 46.6iT - 3.48e3T^{2} \)
61 \( 1 - 40.8T + 3.72e3T^{2} \)
67 \( 1 + 35.0iT - 4.48e3T^{2} \)
71 \( 1 - 106. iT - 5.04e3T^{2} \)
73 \( 1 - 113.T + 5.32e3T^{2} \)
79 \( 1 + 116. iT - 6.24e3T^{2} \)
83 \( 1 - 48.3T + 6.88e3T^{2} \)
89 \( 1 + 24.3iT - 7.92e3T^{2} \)
97 \( 1 - 85.7iT - 9.40e3T^{2} \)
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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

−8.857272268811529523860347251343, −8.072577681817261271939081199175, −6.99726229839596387639336106604, −6.34316103196256856224650940751, −5.68943499743915207619671521560, −4.78605795821780611442010029137, −4.16372602715657169540439255275, −2.66055109320427764675929268848, −1.89064388907210106665467096653, −1.19249490526992616656180289628, 0.931659640979680633698134755361, 1.90585856966171301300908198256, 2.52061445861211431253426483455, 3.87756590438062004054749438648, 4.97122134866100145298661423672, 5.35428201697570716546440266195, 6.37022041741813644884822148696, 6.86151310436424171961141490059, 7.937507247683915931291990236819, 8.868590465834131057428806268158

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