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Properties

Label 1-6048-6048.2531-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.999 + 0.0311i$
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

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

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.999 + 0.0311i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (2531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.999 + 0.0311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.777710504 + 0.02770205301i\)
\(L(\frac12)\) \(\approx\) \(1.777710504 + 0.02770205301i\)
\(L(1)\) \(\approx\) \(1.082566827 - 0.08766490916i\)
\(L(1)\) \(\approx\) \(1.082566827 - 0.08766490916i\)

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

−17.82892600171734431295470746471, −16.857507146836706470877343661137, −16.43316949708857524474313769561, −15.80078469256757138182897429908, −14.96158344550908198798677627215, −14.37001651113593379844041601763, −14.01755948223505111523346538095, −13.20781536026854691258068156837, −12.07087955818941090912844427384, −11.87635259586721111036580877881, −11.13264180621689966200125744029, −10.550801342174926967688941446228, −9.644686092798005645157148133269, −9.14002020033424445862495846811, −8.26853243235393298255583669173, −7.54635168789810301582085612446, −6.99400735838941892261565351940, −6.23082451865626883008899421653, −5.70179777686467011575097711386, −4.44328813366936149341471875073, −4.09221840861862173116965803352, −3.15492753871341472191783172601, −2.63108333239287435393322708112, −1.55198779197373815748414922642, −0.6157848094478305719746391351, 0.8338567544798572434848781926, 1.35198053598783549923722048761, 2.41760541735376416848590746609, 3.458778140912688508601771546553, 3.95614907432030565083661625879, 4.6857161248579955537246707779, 5.70048115866915539884900671140, 5.843208382641119318086710631576, 7.25408991929296686147145599464, 7.660054358840528826206847564037, 8.199080980014944232189920006131, 9.169921718676052183225334520607, 9.63398150297048837577443848272, 10.30979963515726083865657834594, 11.25499444150154955145232937733, 11.977110476598768532827317159920, 12.31378075047692372482881985462, 12.96327354022655525011968107183, 13.82483901122446562850857900927, 14.41727133739655587875596984576, 15.303980542214764219235918231246, 15.60900840989101424331168506605, 16.51146856554740393225165407906, 16.98866065393511985117480983743, 17.56409950039418084793360976705

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