↑

Properties

Label 1-3549-3549.44-r0-0-0
Degree $1$
Conductor $3549$
Sign $-0.640 + 0.767i$
Analytic cond. $16.4814$
Root an. cond. $16.4814$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.534 − 0.845i)2-s + (−0.428 − 0.903i)4-s + (0.160 + 0.987i)5-s + (−0.992 − 0.120i)8-s + (0.919 + 0.391i)10-s + (0.534 + 0.845i)11-s + (−0.632 + 0.774i)16-s + (0.799 + 0.600i)17-s + (−0.866 + 0.5i)19-s + (0.822 − 0.568i)20-s + 22-s + (−0.5 − 0.866i)23-s + (−0.948 + 0.316i)25-s + (−0.885 + 0.464i)29-s + (−0.979 + 0.200i)31-s + (0.316 + 0.948i)32-s + ⋯
L(s)  = 1  + (0.534 − 0.845i)2-s + (−0.428 − 0.903i)4-s + (0.160 + 0.987i)5-s + (−0.992 − 0.120i)8-s + (0.919 + 0.391i)10-s + (0.534 + 0.845i)11-s + (−0.632 + 0.774i)16-s + (0.799 + 0.600i)17-s + (−0.866 + 0.5i)19-s + (0.822 − 0.568i)20-s + 22-s + (−0.5 − 0.866i)23-s + (−0.948 + 0.316i)25-s + (−0.885 + 0.464i)29-s + (−0.979 + 0.200i)31-s + (0.316 + 0.948i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $-0.640 + 0.767i$
Analytic conductor: \(16.4814\)
Root analytic conductor: \(16.4814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (0:\ ),\ -0.640 + 0.767i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1678507884 + 0.3586827403i\)
\(L(\frac12)\) \(\approx\) \(0.1678507884 + 0.3586827403i\)
\(L(1)\) \(\approx\) \(1.042891807 - 0.2248374767i\)
\(L(1)\) \(\approx\) \(1.042891807 - 0.2248374767i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.534 - 0.845i)T \)
5 \( 1 + (0.160 + 0.987i)T \)
11 \( 1 + (0.534 + 0.845i)T \)
17 \( 1 + (0.799 + 0.600i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.885 + 0.464i)T \)
31 \( 1 + (-0.979 + 0.200i)T \)
37 \( 1 + (-0.316 + 0.948i)T \)
41 \( 1 + (-0.239 - 0.970i)T \)
43 \( 1 + (0.748 + 0.663i)T \)
47 \( 1 + (-0.903 - 0.428i)T \)
53 \( 1 + (-0.799 - 0.600i)T \)
59 \( 1 + (-0.774 + 0.632i)T \)
61 \( 1 + (0.799 - 0.600i)T \)
67 \( 1 + (0.0804 + 0.996i)T \)
71 \( 1 + (-0.239 - 0.970i)T \)
73 \( 1 + (-0.534 - 0.845i)T \)
79 \( 1 + (0.428 - 0.903i)T \)
83 \( 1 + (0.239 - 0.970i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.935 + 0.354i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.325521688356441083225123140456, −17.44088112039130658645948996981, −16.99428649510683128111195813400, −16.352487875293735737652679684742, −15.87387349846821627361665181642, −15.05763038305437648366062203957, −14.23285529514509224929279969560, −13.7570146812719231427468662543, −12.98214045083366154192626849198, −12.500076702452537483517007566378, −11.65622217001632550861997447957, −11.069236393537842429224958249589, −9.6372264141474700343465987885, −9.26296041203352375074558904732, −8.47479795843094567538259212518, −7.85257515054398739737352411916, −7.10506375037603578818204317569, −6.1018004067527125038418767969, −5.63565568526892356103409588434, −4.94803025907265895246519057777, −4.04422237683349045778956248776, −3.52983450041506101574423350158, −2.40681089472054881761453672859, −1.27336477032769178156176332374, −0.08917535296297093225259790462, 1.61470193854033549955089062499, 1.94749370057907121659455399072, 2.99900719323639531749190047318, 3.68942076990385318342533003591, 4.31334155671455524815297953934, 5.294795918971297691357853276952, 6.11478313917929573192189638990, 6.65592576742834811347791334546, 7.55506016615266150594343654477, 8.54454034829385737844578130319, 9.44842904971358154702354753129, 10.10826400283797031999103755168, 10.6021762179358363786287154089, 11.26581630518286037521705570871, 12.15729752665350783286982528864, 12.56926127623144473747369914442, 13.38638036323584519462865715592, 14.27951091421733356370556719, 14.754293629616831354674517394088, 15.01311092289144168132255297279, 16.15546402116196083375715902755, 17.076773145376498704359871911, 17.814990611658271735102818957925, 18.44927775940333332311731668034, 19.08613265024252862705303088067

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