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Properties

Label 1-6048-6048.3803-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.0311 - 0.999i$
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  + (0.573 − 0.819i)5-s + (−0.819 + 0.573i)11-s + (0.0871 − 0.996i)13-s + 17-s + (−0.707 + 0.707i)19-s + (−0.642 − 0.766i)23-s + (−0.342 − 0.939i)25-s + (0.996 − 0.0871i)29-s + (0.939 + 0.342i)31-s + (−0.965 + 0.258i)37-s + (0.642 + 0.766i)41-s + (−0.906 − 0.422i)43-s + (0.939 − 0.342i)47-s + (−0.965 + 0.258i)53-s + i·55-s + ⋯
L(s)  = 1  + (0.573 − 0.819i)5-s + (−0.819 + 0.573i)11-s + (0.0871 − 0.996i)13-s + 17-s + (−0.707 + 0.707i)19-s + (−0.642 − 0.766i)23-s + (−0.342 − 0.939i)25-s + (0.996 − 0.0871i)29-s + (0.939 + 0.342i)31-s + (−0.965 + 0.258i)37-s + (0.642 + 0.766i)41-s + (−0.906 − 0.422i)43-s + (0.939 − 0.342i)47-s + (−0.965 + 0.258i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0311 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0311 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.0311 - 0.999i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (3803, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.0311 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.174365237 - 1.211544861i\)
\(L(\frac12)\) \(\approx\) \(1.174365237 - 1.211544861i\)
\(L(1)\) \(\approx\) \(1.086888989 - 0.2672630615i\)
\(L(1)\) \(\approx\) \(1.086888989 - 0.2672630615i\)

Euler product

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

−17.78049685411299189432456114541, −17.38668691587093176738209283265, −16.54427116394644853888285641160, −15.86658717727401389555091393826, −15.30539365557771843891570374767, −14.4540950186013456564028519847, −13.88295747460492335329383352583, −13.54826678389859950996051842989, −12.63190324919574010531293106818, −11.86222923380758149037276199176, −11.19438120481076533341056767344, −10.568374600553960997473577460225, −9.97412041720736575408581836310, −9.341728151564005043377507994044, −8.47751079739809313834787990600, −7.83318771119488461466470518873, −6.98577333124304645232712644355, −6.4496623225587724046644033112, −5.712527288300690684104781484273, −5.09044117388956320013749704808, −4.10433815461889231419438286364, −3.34306812363227989962997758666, −2.58195810144778214384040862211, −1.99473566337988124901854576871, −0.95159991683504407015672233015, 0.46864963260918734284351879644, 1.360237587481062662810642770989, 2.205806625125172504836652714722, 2.912486818113503795973791406674, 3.866192837194843825421388661111, 4.765647355358662360271215845922, 5.22292802473932068479702682314, 5.97066598152786422207460180429, 6.59717283531244882316838898343, 7.70440751849603205267503231082, 8.22049302135308029826516806028, 8.67182210431228213873082120246, 9.840173241598439179001655777354, 10.14104359311651009752320173926, 10.62015887563899600431469081933, 11.86195249838568392784823154819, 12.43589039917383183709856532176, 12.78527046900032615901265390541, 13.56695972657469675494092190478, 14.21947902551835129964027232196, 14.8976416353316734975676775708, 15.77736136716401886730690185304, 16.107342318060034206157814072905, 17.04609163809254229708525329925, 17.42136797552729879857655427015

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