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Properties

Label 2-2736-76.75-c1-0-26
Degree $2$
Conductor $2736$
Sign $1$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.44i·11-s + 4.35·19-s − 7.34i·23-s − 5·25-s − 10.6i·29-s + 8.71·31-s + 10.6i·41-s + 12.2i·47-s + 7·49-s + 10.6i·53-s + 4·61-s + 8.71·67-s − 8·73-s + 17.4·79-s − 17.1i·83-s + ⋯
L(s)  = 1  + 0.738i·11-s + 1.00·19-s − 1.53i·23-s − 25-s − 1.98i·29-s + 1.56·31-s + 1.66i·41-s + 1.78i·47-s + 49-s + 1.46i·53-s + 0.512·61-s + 1.06·67-s − 0.936·73-s + 1.96·79-s − 1.88i·83-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.815136003\)
\(L(\frac12)\) \(\approx\) \(1.815136003\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.35T \)
good5 \( 1 + 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 2.44iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
23 \( 1 + 7.34iT - 23T^{2} \)
29 \( 1 + 10.6iT - 29T^{2} \)
31 \( 1 - 8.71T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 10.6iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 12.2iT - 47T^{2} \)
53 \( 1 - 10.6iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 - 8.71T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 8T + 73T^{2} \)
79 \( 1 - 17.4T + 79T^{2} \)
83 \( 1 + 17.1iT - 83T^{2} \)
89 \( 1 + 10.6iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−8.810956086850532206342820881662, −7.931595683593498849457518172275, −7.49489567655296665223259594414, −6.38705163435327954827334765663, −5.94166324995625963726261382647, −4.67062556177128232334966462633, −4.30660657577960647378903709756, −3.00292553853139417312288190301, −2.19858349996968391383117118224, −0.838750504355816505044282296078, 0.865023383928781860127734522779, 2.07347935549387926473155730649, 3.31531652268793683140684995852, 3.81307744950846888936935066577, 5.23698242596313556070537880208, 5.49422116352070136444732504479, 6.63478122739025774953993269311, 7.28654050907124160503220030161, 8.113377213538805520325791473938, 8.794380527467231823114581818586

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