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Properties

Label 2-3549-273.23-c0-0-4
Degree $2$
Conductor $3549$
Sign $0.325 + 0.945i$
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  + (0.5 − 0.866i)3-s − 4-s + (−0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)12-s + 16-s + (1.5 − 0.866i)19-s + (0.499 + 0.866i)21-s + (0.5 + 0.866i)25-s − 0.999·27-s + (0.5 − 0.866i)28-s + (0.499 + 0.866i)36-s − 1.73i·37-s + (0.5 − 0.866i)43-s + (0.5 − 0.866i)48-s + (−0.499 − 0.866i)49-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s − 4-s + (−0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)12-s + 16-s + (1.5 − 0.866i)19-s + (0.499 + 0.866i)21-s + (0.5 + 0.866i)25-s − 0.999·27-s + (0.5 − 0.866i)28-s + (0.499 + 0.866i)36-s − 1.73i·37-s + (0.5 − 0.866i)43-s + (0.5 − 0.866i)48-s + (−0.499 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.325 + 0.945i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.325 + 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.069127240\)
\(L(\frac12)\) \(\approx\) \(1.069127240\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 \)
good2 \( 1 + T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 1.73iT - T^{2} \)
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 \( 1 + T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.798604350377011364279683819202, −7.81962902786636663733582884373, −7.30885921689551617158836247259, −6.37061479749404294138765012547, −5.55572524722088422683861845233, −4.98354525496135353660427891307, −3.66740600966906727831947129189, −3.10856142378339358710100891357, −2.07311520215323309385635241535, −0.73490870380168569995911126824, 1.13611615628620398520755525226, 2.85952694218353170898065335811, 3.53489197687292661345869453118, 4.23274659068139244325130214524, 4.90789340138444896675404112560, 5.67131222867305439871293320242, 6.67243489639366663344820274814, 7.77447111930962287677622747064, 8.126340765702976520582750505796, 9.031921035340835366122549819538

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