Properties

Label 1-3549-3549.3461-r0-0-0
Degree $1$
Conductor $3549$
Sign $0.812 - 0.583i$
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  + (0.996 + 0.0804i)2-s + (0.987 + 0.160i)4-s + (0.948 − 0.316i)5-s + (0.970 + 0.239i)8-s + (0.970 − 0.239i)10-s + (−0.568 − 0.822i)11-s + (0.948 + 0.316i)16-s + (0.692 − 0.721i)17-s − 19-s + (0.987 − 0.160i)20-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (0.799 − 0.600i)25-s + (−0.428 − 0.903i)29-s + (0.919 − 0.391i)31-s + (0.919 + 0.391i)32-s + ⋯
L(s)  = 1  + (0.996 + 0.0804i)2-s + (0.987 + 0.160i)4-s + (0.948 − 0.316i)5-s + (0.970 + 0.239i)8-s + (0.970 − 0.239i)10-s + (−0.568 − 0.822i)11-s + (0.948 + 0.316i)16-s + (0.692 − 0.721i)17-s − 19-s + (0.987 − 0.160i)20-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (0.799 − 0.600i)25-s + (−0.428 − 0.903i)29-s + (0.919 − 0.391i)31-s + (0.919 + 0.391i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.812 - 0.583i$
Analytic conductor: \(16.4814\)
Root analytic conductor: \(16.4814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (3461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (0:\ ),\ 0.812 - 0.583i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.147443746 - 1.335444917i\)
\(L(\frac12)\) \(\approx\) \(4.147443746 - 1.335444917i\)
\(L(1)\) \(\approx\) \(2.340352860 - 0.2516942853i\)
\(L(1)\) \(\approx\) \(2.340352860 - 0.2516942853i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.996 + 0.0804i)T \)
5 \( 1 + (0.948 - 0.316i)T \)
11 \( 1 + (-0.568 - 0.822i)T \)
17 \( 1 + (0.692 - 0.721i)T \)
19 \( 1 - T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.428 - 0.903i)T \)
31 \( 1 + (0.919 - 0.391i)T \)
37 \( 1 + (-0.919 + 0.391i)T \)
41 \( 1 + (-0.0402 + 0.999i)T \)
43 \( 1 + (0.799 - 0.600i)T \)
47 \( 1 + (-0.632 - 0.774i)T \)
53 \( 1 + (-0.278 - 0.960i)T \)
59 \( 1 + (0.948 - 0.316i)T \)
61 \( 1 + (0.970 - 0.239i)T \)
67 \( 1 + (-0.354 + 0.935i)T \)
71 \( 1 + (0.845 + 0.534i)T \)
73 \( 1 + (-0.428 + 0.903i)T \)
79 \( 1 + (-0.632 - 0.774i)T \)
83 \( 1 + (0.885 + 0.464i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.948 - 0.316i)T \)
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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

−18.97850477990249940647531520760, −18.028419062545173728375747708680, −17.31673967077468705507515841594, −16.73451830890433386678310901741, −15.890686401588746472265203201220, −15.08289095476196108266054560283, −14.59522705743342913985045242015, −14.036268813733741330071924726544, −13.180841870861908514343283713101, −12.61217484006077984809894712082, −12.23091484221417292660786449975, −10.920982631980997474124148665590, −10.58318874578153593238567659712, −9.988334766120334398313894063409, −9.019661394101526520599748990555, −8.046477797580248515532405648830, −7.14801250812386890236345168933, −6.56207162725689002470762710316, −5.8515391635111727674146982979, −5.13535624444507638912206705214, −4.48255391797718496564222520795, −3.52339459502519335552651724502, −2.665118421309599522348465547549, −2.06629461452598926578344797613, −1.25980729617421503357426864100, 0.85936092653333425445079929321, 1.89183598520204958826877253117, 2.62468617744149927914562610005, 3.34825625416865408369489197894, 4.2932680436109488075835836187, 5.21482884291917953534347080301, 5.56761452759370657399472219278, 6.35299738268229380154156952801, 7.05309651931591358747412926732, 8.03632061916750867019612641304, 8.64053635400654569110962802928, 9.80307100999736222684437607923, 10.22224710307727178576923682456, 11.23689686279071715323535072937, 11.68361342929239991002915290381, 12.72043959243427781567420982686, 13.19440619056114363451333126283, 13.73696915825856880524511524313, 14.34609653725773378898121600922, 15.13478217122071943025931169656, 15.859305585402586850549884956753, 16.524631638536305829605682993023, 17.13377304301558813870485744999, 17.752330095679214520630782915292, 18.919949843282807828147481483084

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