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Properties

Label 2-3549-273.179-c0-0-3
Degree $2$
Conductor $3549$
Sign $0.746 - 0.665i$
Analytic cond. $1.77118$
Root an. cond. $1.33085$
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  + 3-s + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)7-s + 9-s + (0.5 + 0.866i)12-s + (−0.499 + 0.866i)16-s + i·19-s + (−0.866 − 0.5i)21-s + (0.5 − 0.866i)25-s + 27-s − 0.999i·28-s + (1.73 + i)31-s + (0.5 + 0.866i)36-s + (0.866 + 0.5i)37-s + (−0.5 + 0.866i)43-s + ⋯
L(s)  = 1  + 3-s + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)7-s + 9-s + (0.5 + 0.866i)12-s + (−0.499 + 0.866i)16-s + i·19-s + (−0.866 − 0.5i)21-s + (0.5 − 0.866i)25-s + 27-s − 0.999i·28-s + (1.73 + i)31-s + (0.5 + 0.866i)36-s + (0.866 + 0.5i)37-s + (−0.5 + 0.866i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.746 - 0.665i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (1544, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.746 - 0.665i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.935628531\)
\(L(\frac12)\) \(\approx\) \(1.935628531\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + 2iT - T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
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

−8.644127809115248737171812593705, −8.059554450740223429182290489657, −7.53626654274057198144173082526, −6.59999241918681510899941237626, −6.30943569768828270496703856253, −4.68840220931324339090616997439, −4.01253049385655196568964563247, −3.14141736534737043264376439409, −2.73761401666886460397118467676, −1.47408669519642256485814320264, 1.10542088073475575595822420804, 2.42759083596634540983900877413, 2.78059953179233360832616056212, 3.90335818972712046814950259395, 4.87413102164184817888939734930, 5.73491589312268813654177033296, 6.59380962106653411126765481443, 7.04097011270165066588540874384, 7.943966334017700608655122584777, 8.838940976519122052088952844292

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