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Properties

Label 2-3549-21.2-c0-0-4
Degree $2$
Conductor $3549$
Sign $0.997 - 0.0633i$
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  + (0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.866 + 0.5i)5-s − 0.999·6-s + (0.5 − 0.866i)7-s − i·8-s + (0.499 − 0.866i)9-s + (0.499 + 0.866i)10-s + (0.866 − 0.499i)14-s − 0.999·15-s + (0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + (0.866 − 0.499i)18-s + 0.999i·21-s + (0.866 + 0.5i)23-s + (0.500 + 0.866i)24-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.866 + 0.5i)5-s − 0.999·6-s + (0.5 − 0.866i)7-s − i·8-s + (0.499 − 0.866i)9-s + (0.499 + 0.866i)10-s + (0.866 − 0.499i)14-s − 0.999·15-s + (0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + (0.866 − 0.499i)18-s + 0.999i·21-s + (0.866 + 0.5i)23-s + (0.500 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.997 - 0.0633i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.997 - 0.0633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.745655844\)
\(L(\frac12)\) \(\approx\) \(1.745655844\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + iT - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - iT - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−8.981654196578344938069948324397, −7.58711698941694326817981893621, −6.99529117170494004340601762642, −6.25736525739070097178661276442, −5.75498117445487873371770651125, −5.08736872898104167330005748207, −4.15644397995543290419976810357, −3.86824556852502356809292992330, −2.32622147804246965531194554793, −0.944096726641429801040237829216, 1.44005862046710614988005490001, 2.22168550145716425123315655499, 3.05918275912218564190744286965, 4.53438768286729147973005780310, 4.91216716290624993071109739580, 5.54347531447502642399740918064, 6.17714019574574606602296400339, 7.08702646211805955442728500833, 8.006901599910303953208350087610, 8.867118623682439267179453959178

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