Properties

Label 2-2736-171.94-c0-0-1
Degree $2$
Conductor $2736$
Sign $0.984 - 0.173i$
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  i·3-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s − 9-s + (−0.5 − 0.866i)11-s + (0.866 + 0.5i)13-s + (0.866 + 0.5i)15-s i·19-s + (0.866 − 0.5i)21-s + (−0.5 + 0.866i)23-s + i·27-s + (0.866 − 0.5i)29-s + (0.866 + 0.5i)31-s + (−0.866 + 0.5i)33-s − 0.999·35-s + ⋯
L(s)  = 1  i·3-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s − 9-s + (−0.5 − 0.866i)11-s + (0.866 + 0.5i)13-s + (0.866 + 0.5i)15-s i·19-s + (0.866 − 0.5i)21-s + (−0.5 + 0.866i)23-s + i·27-s + (0.866 − 0.5i)29-s + (0.866 + 0.5i)31-s + (−0.866 + 0.5i)33-s − 0.999·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1633, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :0),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.128399057\)
\(L(\frac12)\) \(\approx\) \(1.128399057\)
\(L(1)\) 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 \( 1 + iT \)
19 \( 1 + iT \)
good5 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.625689496750483960989559394930, −8.345217973022448612560413197203, −7.50021725428402051919473591594, −6.78928803717125547005704235005, −6.04063760208907748150002656685, −5.45144004586959029827187297512, −4.21885901724040966352168439358, −2.99401969470734593869664762347, −2.56575915008814794383861386600, −1.21263607528423496615475865281, 0.866814079244941949104423492544, 2.40186543378925588299397966844, 3.75510182771274784213227387490, 4.21195935534359479455530414920, 4.90894974542624517736927906414, 5.65111771491815810474994982653, 6.68836878897320700480019207427, 7.85787638298576248819930082371, 8.205078541238790388329651773956, 8.880399765641498956628931968453

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