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Properties

Label 1-6048-6048.1453-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.476 + 0.879i$
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  + (0.819 + 0.573i)5-s + (0.573 + 0.819i)11-s + (−0.573 + 0.819i)13-s + (0.5 − 0.866i)17-s + (0.258 − 0.965i)19-s + (0.642 + 0.766i)23-s + (0.342 + 0.939i)25-s + (−0.819 + 0.573i)29-s + (0.173 + 0.984i)31-s + (0.707 − 0.707i)37-s + (−0.984 + 0.173i)41-s + (−0.996 + 0.0871i)43-s + (−0.173 + 0.984i)47-s + (0.965 + 0.258i)53-s + i·55-s + ⋯
L(s)  = 1  + (0.819 + 0.573i)5-s + (0.573 + 0.819i)11-s + (−0.573 + 0.819i)13-s + (0.5 − 0.866i)17-s + (0.258 − 0.965i)19-s + (0.642 + 0.766i)23-s + (0.342 + 0.939i)25-s + (−0.819 + 0.573i)29-s + (0.173 + 0.984i)31-s + (0.707 − 0.707i)37-s + (−0.984 + 0.173i)41-s + (−0.996 + 0.0871i)43-s + (−0.173 + 0.984i)47-s + (0.965 + 0.258i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.476 + 0.879i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (1453, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.476 + 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9185708689 + 1.542514188i\)
\(L(\frac12)\) \(\approx\) \(0.9185708689 + 1.542514188i\)
\(L(1)\) \(\approx\) \(1.149355137 + 0.3544502500i\)
\(L(1)\) \(\approx\) \(1.149355137 + 0.3544502500i\)

Euler product

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

−17.12570200075492107216641975706, −16.88233016938326361741364173065, −16.57739783537119335497447577838, −15.39825105913354692697119745136, −14.826915217600677849749312924887, −14.26568721511065387699218900201, −13.32048226544946883706206073425, −13.13014369518876027871580993111, −12.173138037627625983617535637369, −11.74507929232944953646205200291, −10.69170894807863678118943206461, −10.14228746417023531544787122648, −9.59489412607846286915832865739, −8.75766321871950108983689375136, −8.21943180000340849879336944047, −7.55254377520475073929128491772, −6.42691300023451436445470364939, −5.98195001131314298882341680607, −5.35549869143918183818378038075, −4.60673731198501552917837018772, −3.68050886536381664793987834398, −3.01691177106905111522193570134, −2.02515581594235585959620420489, −1.34894253336071485309325717511, −0.42969647222331915840141664966, 1.24739227036423025955401077980, 1.81556713020827336397163814891, 2.75397695794355484070077930590, 3.262376564964519746691756809483, 4.40387213916543657646500207877, 4.99762147736202738833158595389, 5.69710222934101319870489602996, 6.63196791149103152654973215998, 7.148364573195549966281024805, 7.51829743149044933178397799113, 9.008496894420806182905754659407, 9.2004467018500146847183830097, 9.895280450243249253921070658061, 10.54509720745695692686121841970, 11.46201959223993640201110945911, 11.81824872060268417921093584519, 12.73814393098678673241867371122, 13.432353926098848890989075661720, 14.01762095486989224778705935978, 14.665233287259123927171471899603, 15.08751585888288922401745033810, 15.989860140505090025268717597769, 16.81634278132001982802437765626, 17.19799961986254976740896742958, 18.04093143355218745772488861547

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