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Properties

Label 1-3549-3549.11-r0-0-0
Degree $1$
Conductor $3549$
Sign $-0.344 + 0.938i$
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  + (−0.999 + 0.0402i)2-s + (0.996 − 0.0804i)4-s + (0.160 − 0.987i)5-s + (−0.992 + 0.120i)8-s + (−0.120 + 0.992i)10-s + (0.464 + 0.885i)11-s + (0.987 − 0.160i)16-s + (−0.919 − 0.391i)17-s + i·19-s + (0.0804 − 0.996i)20-s + (−0.5 − 0.866i)22-s + (−0.5 − 0.866i)23-s + (−0.948 − 0.316i)25-s + (0.845 − 0.534i)29-s + (−0.979 − 0.200i)31-s + (−0.979 + 0.200i)32-s + ⋯
L(s)  = 1  + (−0.999 + 0.0402i)2-s + (0.996 − 0.0804i)4-s + (0.160 − 0.987i)5-s + (−0.992 + 0.120i)8-s + (−0.120 + 0.992i)10-s + (0.464 + 0.885i)11-s + (0.987 − 0.160i)16-s + (−0.919 − 0.391i)17-s + i·19-s + (0.0804 − 0.996i)20-s + (−0.5 − 0.866i)22-s + (−0.5 − 0.866i)23-s + (−0.948 − 0.316i)25-s + (0.845 − 0.534i)29-s + (−0.979 − 0.200i)31-s + (−0.979 + 0.200i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $-0.344 + 0.938i$
Analytic conductor: \(16.4814\)
Root analytic conductor: \(16.4814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (0:\ ),\ -0.344 + 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2032505658 + 0.2911137700i\)
\(L(\frac12)\) \(\approx\) \(0.2032505658 + 0.2911137700i\)
\(L(1)\) \(\approx\) \(0.6129948148 - 0.03924501476i\)
\(L(1)\) \(\approx\) \(0.6129948148 - 0.03924501476i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.999 + 0.0402i)T \)
5 \( 1 + (0.160 - 0.987i)T \)
11 \( 1 + (0.464 + 0.885i)T \)
17 \( 1 + (-0.919 - 0.391i)T \)
19 \( 1 + iT \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.845 - 0.534i)T \)
31 \( 1 + (-0.979 - 0.200i)T \)
37 \( 1 + (0.979 + 0.200i)T \)
41 \( 1 + (-0.721 - 0.692i)T \)
43 \( 1 + (-0.948 - 0.316i)T \)
47 \( 1 + (-0.903 + 0.428i)T \)
53 \( 1 + (-0.799 + 0.600i)T \)
59 \( 1 + (-0.160 + 0.987i)T \)
61 \( 1 + (0.120 - 0.992i)T \)
67 \( 1 + (0.822 + 0.568i)T \)
71 \( 1 + (0.960 - 0.278i)T \)
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 \( 1 + (0.160 + 0.987i)T \)
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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

−18.39716068977697982113589672683, −17.872610385791183967717447907367, −17.39139065444765188176614580964, −16.47354403485275780289048418572, −15.913134686664865568295698453871, −15.11431456900878361524709427534, −14.61123536901664658579860988089, −13.69438525988120771426691498857, −13.02379142103753292590526338907, −11.89793489327572508598904824018, −11.19374847980226754619958575488, −10.99246252366946130071947884796, −10.0365891057289051052723152357, −9.4314226737060531503102323391, −8.664107871047197231537024976595, −8.00460022750973541200948776334, −7.12082679103600480724571999488, −6.51264319147561152758051451932, −6.0495935223825915590829010250, −4.93413121192898680319922108498, −3.60511484229854585865796384895, −3.13344851867971188227645934235, −2.19449211684050627714236690689, −1.45565950246992623583790162070, −0.15199898210538983752616651008, 1.02248772769720743912288841694, 1.85005998188294755953312733584, 2.44739945536920145507803413162, 3.753428434339836746904392836573, 4.54231709534508520459477130373, 5.39893181793950628595849804163, 6.329230140727323937839956863784, 6.86082988610162625034968962342, 7.92361231044522923638603463331, 8.31912760807163324861450580552, 9.20287024265851035290873380411, 9.65999315709143354073343021424, 10.328196500231980192589044138778, 11.22736893134633001482256729997, 12.04531811734573769633689118684, 12.42105170128784886816352234150, 13.27559214107505459629180063508, 14.2497472674884525506794458981, 14.96458230497666482801114948761, 15.76439522939904875826902627376, 16.300394628955504268738264943054, 17.002601682044623214315525709855, 17.45736426253267332272452477282, 18.23083854484269240827576024409, 18.76134937943968814577147589907

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