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Properties

Label 1-3549-3549.2420-r1-0-0
Degree $1$
Conductor $3549$
Sign $0.351 + 0.936i$
Analytic cond. $381.393$
Root an. cond. $381.393$
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  + (−0.935 − 0.354i)2-s + (0.748 + 0.663i)4-s + (0.391 + 0.919i)5-s + (−0.464 − 0.885i)8-s + (−0.0402 − 0.999i)10-s + (0.774 + 0.632i)11-s + (0.120 + 0.992i)16-s + (−0.885 + 0.464i)17-s + (−0.866 − 0.5i)19-s + (−0.316 + 0.948i)20-s + (−0.5 − 0.866i)22-s + 23-s + (−0.692 + 0.721i)25-s + (0.632 + 0.774i)29-s + (0.960 + 0.278i)31-s + (0.239 − 0.970i)32-s + ⋯
L(s)  = 1  + (−0.935 − 0.354i)2-s + (0.748 + 0.663i)4-s + (0.391 + 0.919i)5-s + (−0.464 − 0.885i)8-s + (−0.0402 − 0.999i)10-s + (0.774 + 0.632i)11-s + (0.120 + 0.992i)16-s + (−0.885 + 0.464i)17-s + (−0.866 − 0.5i)19-s + (−0.316 + 0.948i)20-s + (−0.5 − 0.866i)22-s + 23-s + (−0.692 + 0.721i)25-s + (0.632 + 0.774i)29-s + (0.960 + 0.278i)31-s + (0.239 − 0.970i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.351 + 0.936i$
Analytic conductor: \(381.393\)
Root analytic conductor: \(381.393\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2420, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (1:\ ),\ 0.351 + 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.361331311 + 0.9430546111i\)
\(L(\frac12)\) \(\approx\) \(1.361331311 + 0.9430546111i\)
\(L(1)\) \(\approx\) \(0.8180087738 + 0.1257125259i\)
\(L(1)\) \(\approx\) \(0.8180087738 + 0.1257125259i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.935 - 0.354i)T \)
5 \( 1 + (0.391 + 0.919i)T \)
11 \( 1 + (0.774 + 0.632i)T \)
17 \( 1 + (-0.885 + 0.464i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + T \)
29 \( 1 + (0.632 + 0.774i)T \)
31 \( 1 + (0.960 + 0.278i)T \)
37 \( 1 + (-0.239 - 0.970i)T \)
41 \( 1 + (0.0804 + 0.996i)T \)
43 \( 1 + (-0.278 - 0.960i)T \)
47 \( 1 + (-0.316 + 0.948i)T \)
53 \( 1 + (0.0402 - 0.999i)T \)
59 \( 1 + (0.992 + 0.120i)T \)
61 \( 1 + (0.845 - 0.534i)T \)
67 \( 1 + (0.316 - 0.948i)T \)
71 \( 1 + (0.903 + 0.428i)T \)
73 \( 1 + (0.160 - 0.987i)T \)
79 \( 1 + (0.948 + 0.316i)T \)
83 \( 1 + (-0.822 - 0.568i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.600 + 0.799i)T \)
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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

−18.39381858805791727890835786008, −17.4251492486304493360303024109, −17.16534652793034255161201454169, −16.552174460087927226050914609617, −15.81463830236539306012057452399, −15.21877098808794170463846190052, −14.3372824220250406737077665416, −13.642909403636984409289103984299, −12.93081145242732113986298780247, −11.89634675427525588587091216168, −11.48573922539711927720523877120, −10.54589071952169049812399970264, −9.84073765976931469195133937440, −9.166563614038722103850527872447, −8.511886695484253506623373701352, −8.180065743019810039931350591060, −6.93891321783955128667851235215, −6.434916723199386081485079166382, −5.69446117059606439886214196179, −4.86718071065111249483268956386, −4.06334942935525337186584652074, −2.783349573206078175636489778673, −1.98792698060489961175952764238, −1.06559446199623894117450902853, −0.49142900752010096449110096805, 0.72526932379560287915496753621, 1.78373480436937245908833455982, 2.336243711054172355105299803685, 3.18928805588385782804875847193, 3.99074019180542369720105826519, 4.96338317771218710258511130696, 6.323883529562707690035409705890, 6.67348422104933519882065799731, 7.2171971256996721213798529910, 8.2728487955657574739081312139, 8.91641830201501712496221440290, 9.609356904796791300903051720871, 10.284685135871848576585392691298, 10.98500827597771965235756000746, 11.3790691667904899555035688770, 12.3894700311367565499662680543, 12.949853381555296396371937271873, 13.85917021257819464816220572265, 14.79677306912621072693965583835, 15.19379774287493233365977785800, 16.00659235856962117017119123902, 16.93640045955091588748640374530, 17.58757461911777726307898476737, 17.786588291577712785611007286300, 18.70417389386293142407021755515

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