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Properties

Label 2-3549-273.107-c0-0-3
Degree $2$
Conductor $3549$
Sign $0.890 + 0.455i$
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  − i·2-s + (0.866 + 0.5i)3-s + (0.866 + 0.5i)5-s + (0.5 − 0.866i)6-s + (0.5 + 0.866i)7-s − i·8-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)10-s + (0.866 − 0.5i)14-s + (0.499 + 0.866i)15-s − 16-s − i·17-s + (0.866 − 0.499i)18-s + 0.999i·21-s − i·23-s + (0.5 − 0.866i)24-s + ⋯
L(s)  = 1  − i·2-s + (0.866 + 0.5i)3-s + (0.866 + 0.5i)5-s + (0.5 − 0.866i)6-s + (0.5 + 0.866i)7-s − i·8-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)10-s + (0.866 − 0.5i)14-s + (0.499 + 0.866i)15-s − 16-s − i·17-s + (0.866 − 0.499i)18-s + 0.999i·21-s − i·23-s + (0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.890 + 0.455i$
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: \((2,\ 3549,\ (\ :0),\ 0.890 + 0.455i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.354163626\)
\(L(\frac12)\) \(\approx\) \(2.354163626\)
\(L(1)\) 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 \( 1 + iT - T^{2} \)
5 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + iT - T^{2} \)
29 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
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 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + iT - T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + iT - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.991467412249484123187020495466, −8.164161465233552998286858413943, −7.25029354575051837693259463267, −6.50272159461540931910327404944, −5.52457752889568522291791341764, −4.72826686179399511543001592120, −3.72600083861810348519052152335, −2.82727628237077085975365676158, −2.34852067084276385749236677723, −1.64386588921025527059791694418, 1.55626075593631418698067638373, 1.96861395291925751262698103846, 3.35928859827508056602516124980, 4.22445639357641663119973548974, 5.38620608056053054488089033824, 5.79685594964412561324847151107, 6.89762976894661481095198282728, 7.21356198534970528034432074353, 7.969545416473219135866105580918, 8.642301791507376077195996862089

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