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Properties

Label 2-2736-228.227-c0-0-10
Degree $2$
Conductor $2736$
Sign $-0.908 + 0.418i$
Analytic cond. $1.36544$
Root an. cond. $1.16852$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.93i·5-s + i·7-s − 1.93·11-s − 0.517i·17-s − i·19-s − 1.41·23-s − 2.73·25-s + 1.93·35-s − 1.73i·43-s + 0.517·47-s + 3.73i·55-s − 1.73·61-s − 73-s − 1.93i·77-s + 1.41·83-s + ⋯
L(s)  = 1  − 1.93i·5-s + i·7-s − 1.93·11-s − 0.517i·17-s − i·19-s − 1.41·23-s − 2.73·25-s + 1.93·35-s − 1.73i·43-s + 0.517·47-s + 3.73i·55-s − 1.73·61-s − 73-s − 1.93i·77-s + 1.41·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.908 + 0.418i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2735, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :0),\ -0.908 + 0.418i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5789843380\)
\(L(\frac12)\) \(\approx\) \(0.5789843380\)
\(L(1)\) 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 \( 1 + iT \)
good5 \( 1 + 1.93iT - T^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 + 1.93T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 0.517iT - T^{2} \)
23 \( 1 + 1.41T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + 1.73iT - T^{2} \)
47 \( 1 - 0.517T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.73T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.768023651352264485095572631345, −8.035384069926729763254416096496, −7.49932610770339486630728059281, −6.06273402715115050056194350758, −5.32645538800995296924400381764, −5.04559306447479934794994027294, −4.12910019873926478865732751075, −2.71157440457987262398246981670, −1.92800372331885536156797424444, −0.33621524460988506890920204207, 1.97556073725916873941508651807, 2.87296423400372615319595313313, 3.59084606235859465546508087826, 4.47160297415081273664609038226, 5.79389342278027564452531630299, 6.22992941158733707669985663770, 7.24849135833409535718776060620, 7.71415707032317004086231541543, 8.161680858577362267074985390650, 9.735021410767016169660010183776

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