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Properties

Label 2-2736-19.18-c2-0-87
Degree $2$
Conductor $2736$
Sign $-0.982 + 0.188i$
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  + 1.70·5-s + 0.562·7-s − 1.21·11-s + 7.10i·13-s − 3.07·17-s + (18.6 − 3.58i)19-s − 33.1·23-s − 22.0·25-s + 22.6i·29-s − 29.3i·31-s + 0.960·35-s − 23.9i·37-s + 36.6i·41-s + 18.0·43-s − 36.6·47-s + ⋯
L(s)  = 1  + 0.341·5-s + 0.0804·7-s − 0.110·11-s + 0.546i·13-s − 0.181·17-s + (0.982 − 0.188i)19-s − 1.44·23-s − 0.883·25-s + 0.780i·29-s − 0.945i·31-s + 0.0274·35-s − 0.648i·37-s + 0.893i·41-s + 0.419·43-s − 0.779·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.188i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.982 + 0.188i$
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: \((2,\ 2736,\ (\ :1),\ -0.982 + 0.188i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1074764878\)
\(L(\frac12)\) \(\approx\) \(0.1074764878\)
\(L(2)\) 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 \( 1 + (-18.6 + 3.58i)T \)
good5 \( 1 - 1.70T + 25T^{2} \)
7 \( 1 - 0.562T + 49T^{2} \)
11 \( 1 + 1.21T + 121T^{2} \)
13 \( 1 - 7.10iT - 169T^{2} \)
17 \( 1 + 3.07T + 289T^{2} \)
23 \( 1 + 33.1T + 529T^{2} \)
29 \( 1 - 22.6iT - 841T^{2} \)
31 \( 1 + 29.3iT - 961T^{2} \)
37 \( 1 + 23.9iT - 1.36e3T^{2} \)
41 \( 1 - 36.6iT - 1.68e3T^{2} \)
43 \( 1 - 18.0T + 1.84e3T^{2} \)
47 \( 1 + 36.6T + 2.20e3T^{2} \)
53 \( 1 + 1.41iT - 2.80e3T^{2} \)
59 \( 1 + 35.4iT - 3.48e3T^{2} \)
61 \( 1 + 6.68T + 3.72e3T^{2} \)
67 \( 1 + 69.4iT - 4.48e3T^{2} \)
71 \( 1 - 56.3iT - 5.04e3T^{2} \)
73 \( 1 - 25.1T + 5.32e3T^{2} \)
79 \( 1 + 105. iT - 6.24e3T^{2} \)
83 \( 1 + 49.5T + 6.88e3T^{2} \)
89 \( 1 - 100. iT - 7.92e3T^{2} \)
97 \( 1 - 20.1iT - 9.40e3T^{2} \)
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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

−8.150817705907307556404903250024, −7.66861213375522117218702023265, −6.69351482181271902751242036915, −6.00338969560458542210779623757, −5.22514008072072760761140985882, −4.32125059622219209716057155895, −3.46739639277118459400526644478, −2.35128356598769730691497388213, −1.48706289886140357817261188004, −0.02351206185500326424918402142, 1.35341432956112858692009519726, 2.36475949270134176700940790645, 3.36764766496040295382988724234, 4.24916739411658467004984442934, 5.26687024242905059336153011929, 5.86236830803516942232670348490, 6.66514469093822700178231459112, 7.65178959540264438899550309710, 8.125678699814020869753647952490, 9.013795722474016813149544746237

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