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Properties

Label 1-3549-3549.62-r0-0-0
Degree $1$
Conductor $3549$
Sign $0.957 + 0.287i$
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  + (−0.632 − 0.774i)2-s + (−0.200 + 0.979i)4-s + (−0.120 + 0.992i)5-s + (0.885 − 0.464i)8-s + (0.845 − 0.534i)10-s + (0.987 + 0.160i)11-s + (−0.919 − 0.391i)16-s + (−0.845 − 0.534i)17-s + (−0.5 − 0.866i)19-s + (−0.948 − 0.316i)20-s + (−0.5 − 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.970 − 0.239i)25-s + (0.632 + 0.774i)29-s + (−0.970 + 0.239i)31-s + (0.278 + 0.960i)32-s + ⋯
L(s)  = 1  + (−0.632 − 0.774i)2-s + (−0.200 + 0.979i)4-s + (−0.120 + 0.992i)5-s + (0.885 − 0.464i)8-s + (0.845 − 0.534i)10-s + (0.987 + 0.160i)11-s + (−0.919 − 0.391i)16-s + (−0.845 − 0.534i)17-s + (−0.5 − 0.866i)19-s + (−0.948 − 0.316i)20-s + (−0.5 − 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.970 − 0.239i)25-s + (0.632 + 0.774i)29-s + (−0.970 + 0.239i)31-s + (0.278 + 0.960i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.957 + 0.287i$
Analytic conductor: \(16.4814\)
Root analytic conductor: \(16.4814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (0:\ ),\ 0.957 + 0.287i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.057101069 + 0.1550283668i\)
\(L(\frac12)\) \(\approx\) \(1.057101069 + 0.1550283668i\)
\(L(1)\) \(\approx\) \(0.7728995388 - 0.07486283169i\)
\(L(1)\) \(\approx\) \(0.7728995388 - 0.07486283169i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.632 - 0.774i)T \)
5 \( 1 + (-0.120 + 0.992i)T \)
11 \( 1 + (0.987 + 0.160i)T \)
17 \( 1 + (-0.845 - 0.534i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.632 + 0.774i)T \)
31 \( 1 + (-0.970 + 0.239i)T \)
37 \( 1 + (-0.278 + 0.960i)T \)
41 \( 1 + (0.996 - 0.0804i)T \)
43 \( 1 + (0.278 + 0.960i)T \)
47 \( 1 + (0.748 - 0.663i)T \)
53 \( 1 + (-0.885 + 0.464i)T \)
59 \( 1 + (0.919 - 0.391i)T \)
61 \( 1 + (0.0402 + 0.999i)T \)
67 \( 1 + (0.200 + 0.979i)T \)
71 \( 1 + (0.428 - 0.903i)T \)
73 \( 1 + (-0.354 - 0.935i)T \)
79 \( 1 + (-0.748 + 0.663i)T \)
83 \( 1 + (-0.568 + 0.822i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.799 - 0.600i)T \)
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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

−18.78608051492567822155634773150, −17.55324829993408526478958857534, −17.38387738786081360215666956818, −16.728788759029111236276188248124, −15.94923814770204672876370407864, −15.52920962952912408263185620063, −14.59529726107447897269268123353, −14.08297326717415484089258868257, −13.177024365240324970718426074963, −12.57574674944682708306122385462, −11.61904317620052350916763454318, −10.96648582376840285833736645625, −10.088566399828087350723197800894, −9.16860513965918127233077860148, −8.98376988816776227120067494832, −8.11657191036650946196338403060, −7.49341674796820014128007841407, −6.56543524127720093208471338446, −5.90618316370326605595054954805, −5.25407677750989618383720598006, −4.25181091743399469323431031816, −3.82853354916338434003122153505, −2.14465846824033671327830792881, −1.47617194992768460259612526994, −0.549610767175877875372269372057, 0.76403301672961417115176140863, 1.85098777310257882989561442464, 2.63591252417937052565939671315, 3.22613375193849275020356119539, 4.181651442615172068027000044493, 4.74507871348062808056300661986, 6.17521453937595504091789270618, 6.979596100284908142079438176797, 7.22292066134856901195589309370, 8.45110384265457779255829260960, 8.95551504551465820488709211915, 9.68115332502447754467526947424, 10.49732146061044412596942470939, 11.08924445096299429825508804796, 11.526866331451070239151704955950, 12.3845720025319665696831421157, 13.05817986097364943720985366156, 13.92858528915793371312902409297, 14.52888887252193211860227863771, 15.34068848035366475058458920721, 16.12852841897412677790249930456, 16.89741073599812912126050065486, 17.65013795647099086300462958469, 18.0686009227824247522551303511, 18.79659391077341574603600710824

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