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Properties

Label 2-2736-19.18-c2-0-81
Degree $2$
Conductor $2736$
Sign $0.961 + 0.274i$
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  + 4.01·5-s + 10.9·7-s + 9.36·11-s − 15.4i·13-s + 13.8·17-s + (18.2 + 5.22i)19-s + 35.0·23-s − 8.88·25-s + 25.0i·29-s + 20.4i·31-s + 43.8·35-s − 51.6i·37-s − 52.7i·41-s − 23.7·43-s − 4.83·47-s + ⋯
L(s)  = 1  + 0.802·5-s + 1.55·7-s + 0.851·11-s − 1.18i·13-s + 0.812·17-s + (0.961 + 0.274i)19-s + 1.52·23-s − 0.355·25-s + 0.865i·29-s + 0.660i·31-s + 1.25·35-s − 1.39i·37-s − 1.28i·41-s − 0.552·43-s − 0.102·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.961 + 0.274i$
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: \((2,\ 2736,\ (\ :1),\ 0.961 + 0.274i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.739656057\)
\(L(\frac12)\) \(\approx\) \(3.739656057\)
\(L(2)\) 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 \( 1 + (-18.2 - 5.22i)T \)
good5 \( 1 - 4.01T + 25T^{2} \)
7 \( 1 - 10.9T + 49T^{2} \)
11 \( 1 - 9.36T + 121T^{2} \)
13 \( 1 + 15.4iT - 169T^{2} \)
17 \( 1 - 13.8T + 289T^{2} \)
23 \( 1 - 35.0T + 529T^{2} \)
29 \( 1 - 25.0iT - 841T^{2} \)
31 \( 1 - 20.4iT - 961T^{2} \)
37 \( 1 + 51.6iT - 1.36e3T^{2} \)
41 \( 1 + 52.7iT - 1.68e3T^{2} \)
43 \( 1 + 23.7T + 1.84e3T^{2} \)
47 \( 1 + 4.83T + 2.20e3T^{2} \)
53 \( 1 - 70.7iT - 2.80e3T^{2} \)
59 \( 1 - 81.7iT - 3.48e3T^{2} \)
61 \( 1 + 12.6T + 3.72e3T^{2} \)
67 \( 1 + 74.5iT - 4.48e3T^{2} \)
71 \( 1 + 52.2iT - 5.04e3T^{2} \)
73 \( 1 + 119.T + 5.32e3T^{2} \)
79 \( 1 + 118. iT - 6.24e3T^{2} \)
83 \( 1 - 0.221T + 6.88e3T^{2} \)
89 \( 1 - 77.3iT - 7.92e3T^{2} \)
97 \( 1 + 104. iT - 9.40e3T^{2} \)
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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

−8.842458645193966671577938572950, −7.62389453854069913797435776937, −7.42794440348886929625188435970, −6.14801213998515199625269463115, −5.35252091463320025232849737111, −5.02136604393758223246567074830, −3.78223052722557401354671351470, −2.84829407021279241899146337964, −1.61621669151163732167682861734, −1.05066624438220007759474270883, 1.21047699156466606806829703097, 1.68592966361625162296916359264, 2.84409820671548975817261436802, 4.06344897103910891833844746273, 4.85265203516863798606867621574, 5.46502050822114535323760683112, 6.43400851866223372287802918398, 7.12182291785125471247602008273, 8.020572298914415475861304409884, 8.637110445459408359563301264482

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