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Properties

Label 2-3549-3549.2672-c0-0-0
Degree $2$
Conductor $3549$
Sign $0.963 + 0.266i$
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  + (0.822 − 0.568i)3-s + (0.316 + 0.948i)4-s + (−0.0402 − 0.999i)7-s + (0.354 − 0.935i)9-s + (0.799 + 0.600i)12-s + (0.866 + 0.5i)13-s + (−0.799 + 0.600i)16-s + (−0.826 + 0.826i)19-s + (−0.600 − 0.799i)21-s + (0.960 − 0.278i)25-s + (−0.239 − 0.970i)27-s + (0.935 − 0.354i)28-s + (1.99 + 0.0402i)31-s + (0.999 + 0.0402i)36-s + (−1.35 − 0.0273i)37-s + ⋯
L(s)  = 1  + (0.822 − 0.568i)3-s + (0.316 + 0.948i)4-s + (−0.0402 − 0.999i)7-s + (0.354 − 0.935i)9-s + (0.799 + 0.600i)12-s + (0.866 + 0.5i)13-s + (−0.799 + 0.600i)16-s + (−0.826 + 0.826i)19-s + (−0.600 − 0.799i)21-s + (0.960 − 0.278i)25-s + (−0.239 − 0.970i)27-s + (0.935 − 0.354i)28-s + (1.99 + 0.0402i)31-s + (0.999 + 0.0402i)36-s + (−1.35 − 0.0273i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.963 + 0.266i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2672, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.963 + 0.266i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.911296515\)
\(L(\frac12)\) \(\approx\) \(1.911296515\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.822 + 0.568i)T \)
7 \( 1 + (0.0402 + 0.999i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
good2 \( 1 + (-0.316 - 0.948i)T^{2} \)
5 \( 1 + (-0.960 + 0.278i)T^{2} \)
11 \( 1 + (-0.663 - 0.748i)T^{2} \)
17 \( 1 + (0.996 + 0.0804i)T^{2} \)
19 \( 1 + (0.826 - 0.826i)T - iT^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.200 - 0.979i)T^{2} \)
31 \( 1 + (-1.99 - 0.0402i)T + (0.999 + 0.0402i)T^{2} \)
37 \( 1 + (1.35 + 0.0273i)T + (0.999 + 0.0402i)T^{2} \)
41 \( 1 + (0.160 + 0.987i)T^{2} \)
43 \( 1 + (-1.91 + 0.555i)T + (0.845 - 0.534i)T^{2} \)
47 \( 1 + (0.391 + 0.919i)T^{2} \)
53 \( 1 + (-0.428 - 0.903i)T^{2} \)
59 \( 1 + (0.960 - 0.278i)T^{2} \)
61 \( 1 + (-0.881 + 1.67i)T + (-0.568 - 0.822i)T^{2} \)
67 \( 1 + (1.03 + 0.0624i)T + (0.992 + 0.120i)T^{2} \)
71 \( 1 + (-0.774 + 0.632i)T^{2} \)
73 \( 1 + (1.97 - 0.199i)T + (0.979 - 0.200i)T^{2} \)
79 \( 1 + (-0.329 - 1.61i)T + (-0.919 + 0.391i)T^{2} \)
83 \( 1 + (0.935 - 0.354i)T^{2} \)
89 \( 1 + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (-0.0727 + 0.512i)T + (-0.960 - 0.278i)T^{2} \)
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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

−8.438784033145136701054249073323, −8.113573368478096930206566813454, −7.22218059435609989634389449858, −6.73721149951492076351911918867, −6.10752514906036939587709828097, −4.48731488818077669811108597691, −3.92999119051794699870440419583, −3.21947849124319298408663629593, −2.29732658709050404107038956894, −1.23129620687551039570966570268, 1.34496853597974097412554383174, 2.50901971809618592685676326660, 2.96582743065046122537696587368, 4.26785933238880684358675206409, 4.96154671746522183162139919841, 5.76018618235246203771080132910, 6.42647910994277216915062993089, 7.30646513759831288032619390755, 8.342990655859157128148289853148, 8.859634915491680683478044818669

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