Properties

Label 1-3549-3549.131-r0-0-0
Degree $1$
Conductor $3549$
Sign $0.0183 + 0.999i$
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

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0183 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0183 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.0183 + 0.999i$
Analytic conductor: \(16.4814\)
Root analytic conductor: \(16.4814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (0:\ ),\ 0.0183 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.898748305 + 2.846144340i\)
\(L(\frac12)\) \(\approx\) \(2.898748305 + 2.846144340i\)
\(L(1)\) \(\approx\) \(1.976642969 + 0.9541147737i\)
\(L(1)\) \(\approx\) \(1.976642969 + 0.9541147737i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−18.537572044805757843630106892882, −18.110882673798124650861373361084, −16.9670207361541344595773526768, −16.69764013762621059781435778439, −15.48183958988847055519835472298, −14.959704738714494505682502776273, −14.21535612869087743340708620795, −13.789420172550225708493276143111, −13.00136694382888413660818328972, −12.44342041593075824088327238675, −11.58120738684509484380529049794, −11.05335672391863782875972491321, −10.169647212308162665451364415810, −9.46210550871152081996238170392, −9.21035641037773559539472808973, −7.739981298930138936993597576858, −6.89175637917752842453360180976, −6.31087406679350328094614537165, −5.56925664376357774619739544794, −4.83671144826422453479043392204, −4.156130327872209386735935093613, −3.08659186812690104634830078506, −2.538667495546377263767906240735, −1.64245525453453695332991583500, −0.87776853526099349692524460078, 1.34402211700076922439633082132, 1.88997095471600292396860991068, 3.30534934924998542774743624934, 3.43572405500250043627755242052, 4.67480788069297674827542386240, 5.404865064949051736133061434864, 5.94156210047493772925184982171, 6.58875082985475968322857902889, 7.390189787817211895541070993438, 8.25721453932173313010874718585, 8.94557192655435186080969774089, 9.73182085042511739164031270555, 10.55925562154003056188895281908, 11.44386831151279168751475871682, 12.12997632422419995268267456816, 12.77825463702626555287271506280, 13.56210476129657065839897636076, 14.00500163596609666366062037428, 14.673647229970312066545869318448, 15.24676559055149966557557893039, 16.32029413206708469753880101198, 16.79080121621636388034060484505, 17.17261692507212196126022042974, 18.08295638589828974363586820559, 18.754386077069697276917524802693

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