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Properties

Label 1-6045-6045.4334-r0-0-0
Degree $1$
Conductor $6045$
Sign $0.985 + 0.170i$
Analytic cond. $28.0728$
Root an. cond. $28.0728$
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  − i·2-s − 4-s + (0.866 + 0.5i)7-s + i·8-s + (−0.866 + 0.5i)11-s + (0.5 − 0.866i)14-s + 16-s + (0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (0.5 + 0.866i)22-s − 23-s + (−0.866 − 0.5i)28-s − 29-s − i·32-s + (−0.866 − 0.5i)34-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (0.866 + 0.5i)7-s + i·8-s + (−0.866 + 0.5i)11-s + (0.5 − 0.866i)14-s + 16-s + (0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (0.5 + 0.866i)22-s − 23-s + (−0.866 − 0.5i)28-s − 29-s − i·32-s + (−0.866 − 0.5i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6045\)    =    \(3 \cdot 5 \cdot 13 \cdot 31\)
Sign: $0.985 + 0.170i$
Analytic conductor: \(28.0728\)
Root analytic conductor: \(28.0728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6045} (4334, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6045,\ (0:\ ),\ 0.985 + 0.170i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.419065901 + 0.1220035705i\)
\(L(\frac12)\) \(\approx\) \(1.419065901 + 0.1220035705i\)
\(L(1)\) \(\approx\) \(0.9510874438 - 0.2936553194i\)
\(L(1)\) \(\approx\) \(0.9510874438 - 0.2936553194i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
31 \( 1 \)
good2 \( 1 - iT \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 - T \)
29 \( 1 - T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 - iT \)
97 \( 1 + iT \)
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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

−17.49039360324712360381369439422, −17.10155987926466719046944875652, −16.24060213096051657355663175910, −15.88426612852704995042531359257, −14.994546899913938521725651735853, −14.518677548308621463367619779858, −13.876237055402716409021501961820, −13.27488789449751590620247389137, −12.700607009349704180828897312157, −11.720458055117986549716039764688, −11.028471984238924948761475184983, −10.25236107641520245760108347628, −9.70494430285183480373821336418, −8.74519298759638883675468447473, −8.155503355921314694728171511128, −7.64131861333674610176023405867, −7.10748546486487322472471382661, −6.07076218872631007602115535834, −5.52469665083688603461966077139, −4.95945440799762872834536201635, −4.05181697557835215848786403912, −3.55065568189148938245107991794, −2.38905218852710074722204344325, −1.3506527044683018677193388258, −0.44353582305244412645237677153, 0.88968430646102243076402883391, 1.739626540038371984641288752610, 2.41247724963580321275141526239, 3.05923455214386613911268249838, 3.99412190447711547888579277793, 4.71615866258209697283624890488, 5.38116418762493603413176662267, 5.8274576837356872225549870172, 7.21255274084849966453631024755, 7.96926211751744425228371077480, 8.24174789161709128832945252365, 9.4298025256776104266872884140, 9.69107352289769248764770831007, 10.46418377363357793929232213895, 11.38226122881354784251461356830, 11.571132287219012152106906404323, 12.444238548839829538934646344176, 12.91061812083003692208766035367, 13.77877749186010132338430993832, 14.35686764694434915093899310132, 14.857173049892865957386028224208, 15.799935092461027225419128000724, 16.392477196007485054958705445225, 17.4160100237258106161818051523, 17.89988755575811616965167346301

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