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Properties

Label 1-6048-6048.2027-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.911 + 0.411i$
Analytic cond. $28.0867$
Root an. cond. $28.0867$
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  + (0.0871 + 0.996i)5-s + (0.996 + 0.0871i)11-s + (0.422 − 0.906i)13-s + 17-s + (0.707 − 0.707i)19-s + (0.342 + 0.939i)23-s + (−0.984 + 0.173i)25-s + (0.906 − 0.422i)29-s + (−0.173 + 0.984i)31-s + (−0.258 + 0.965i)37-s + (−0.342 − 0.939i)41-s + (0.573 − 0.819i)43-s + (−0.173 − 0.984i)47-s + (−0.258 + 0.965i)53-s + i·55-s + ⋯
L(s)  = 1  + (0.0871 + 0.996i)5-s + (0.996 + 0.0871i)11-s + (0.422 − 0.906i)13-s + 17-s + (0.707 − 0.707i)19-s + (0.342 + 0.939i)23-s + (−0.984 + 0.173i)25-s + (0.906 − 0.422i)29-s + (−0.173 + 0.984i)31-s + (−0.258 + 0.965i)37-s + (−0.342 − 0.939i)41-s + (0.573 − 0.819i)43-s + (−0.173 − 0.984i)47-s + (−0.258 + 0.965i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.911 + 0.411i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (2027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.911 + 0.411i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.323732178 + 0.4999795361i\)
\(L(\frac12)\) \(\approx\) \(2.323732178 + 0.4999795361i\)
\(L(1)\) \(\approx\) \(1.293857729 + 0.1753183248i\)
\(L(1)\) \(\approx\) \(1.293857729 + 0.1753183248i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.0871 + 0.996i)T \)
11 \( 1 + (0.996 + 0.0871i)T \)
13 \( 1 + (0.422 - 0.906i)T \)
17 \( 1 + T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (0.342 + 0.939i)T \)
29 \( 1 + (0.906 - 0.422i)T \)
31 \( 1 + (-0.173 + 0.984i)T \)
37 \( 1 + (-0.258 + 0.965i)T \)
41 \( 1 + (-0.342 - 0.939i)T \)
43 \( 1 + (0.573 - 0.819i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (-0.258 + 0.965i)T \)
59 \( 1 + (0.906 + 0.422i)T \)
61 \( 1 + (0.819 + 0.573i)T \)
67 \( 1 + (-0.996 + 0.0871i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (0.906 - 0.422i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.173 + 0.984i)T \)
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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

−17.59118012830454595472711571283, −16.831368629439767842051330819591, −16.27750996235791482138396442264, −16.13785136596657367913431051232, −14.88305507736180141879970374106, −14.30625288058801280910493212553, −13.86746539381527287763050849636, −12.92844397036232927447847925865, −12.41527270607241595424972038540, −11.76710014928038523401353679542, −11.25875502230533210634253028718, −10.239844704628782885843678408399, −9.481863882715651242089995836606, −9.14565646548444089593072395780, −8.281560812418318694625593332146, −7.79630377277685670832493650130, −6.742988636159070834557542318415, −6.18588606464884195557653812636, −5.419468277370890945605176301758, −4.665233525219031898018462112659, −3.99785843965670246085464716666, −3.355098429756325285156087486535, −2.21885937497599321843248875538, −1.34927764256942817022172429458, −0.865784867306642331102488235589, 0.82878818765138056750323905085, 1.58672297693454043861282883639, 2.65799024097575238620964732928, 3.33957823525439824397131042481, 3.746719886065979603898529369245, 4.91391106144701070517619184363, 5.623965400115504523672470438509, 6.269871686445973186322674749978, 7.1158026383980607512925282390, 7.442292163230969170544849140091, 8.43531155695394806995007268047, 9.09068379717304510351110966848, 9.985526064544787872827609695870, 10.34894505360921091176114701857, 11.14950760813414668118464202234, 11.85258225985798793470079386132, 12.25995197402766018478753806311, 13.44545251419390174731656824973, 13.76827714706341488508059719404, 14.487499548760623062586239791283, 15.167828609356625069812865758730, 15.61826157411431053347345159686, 16.41329950873809373379011169407, 17.343809642625061470095571714757, 17.67168089767822128489231210843

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