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Properties

Label 2-3549-273.107-c0-0-2
Degree $2$
Conductor $3549$
Sign $0.795 + 0.606i$
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 + (0.866 + 0.5i)7-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)12-s + 16-s + (0.866 + 1.5i)19-s + (0.866 − 0.499i)21-s + (−0.5 − 0.866i)25-s − 0.999·27-s + (0.866 + 0.5i)28-s + (−0.499 − 0.866i)36-s − 1.73·37-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 + (0.866 + 0.5i)7-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)12-s + 16-s + (0.866 + 1.5i)19-s + (0.866 − 0.499i)21-s + (−0.5 − 0.866i)25-s − 0.999·27-s + (0.866 + 0.5i)28-s + (−0.499 − 0.866i)36-s − 1.73·37-s + (−0.5 + 0.866i)43-s + (0.5 − 0.866i)48-s + (0.499 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.795 + 0.606i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.795 + 0.606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.117870530\)
\(L(\frac12)\) \(\approx\) \(2.117870530\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T \)
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 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + 1.73T + T^{2} \)
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 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.341842528308810192943256738778, −7.902633802517985344660599191744, −7.36004614322624536984571652335, −6.43724257513246240896657147336, −5.90163661378742136935363834672, −5.07343058786051468287813564611, −3.72640027700614220744431703369, −2.95093374238459842463127869838, −1.96862948860913271434557356564, −1.44758545457071734627694642464, 1.47581233420672987487139748645, 2.46498373550549570187149213201, 3.30505441354777963907830174650, 4.10490824913838260422917459534, 5.13001649403389599936093179948, 5.52867993822262665495069683566, 6.89457867441687247974530042758, 7.30015289833782051475192831177, 8.097402233640984565259955108212, 8.785988489396381686628353237494

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