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Properties

Label 1-6048-6048.5501-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.995 - 0.0943i$
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  + (0.422 − 0.906i)5-s + (0.906 − 0.422i)11-s + (0.0871 + 0.996i)13-s + (0.5 − 0.866i)17-s + (0.258 − 0.965i)19-s + (−0.984 − 0.173i)23-s + (−0.642 − 0.766i)25-s + (−0.996 − 0.0871i)29-s + (−0.173 + 0.984i)31-s + (−0.258 − 0.965i)37-s + (−0.642 + 0.766i)41-s + (−0.906 + 0.422i)43-s + (−0.173 − 0.984i)47-s + (−0.707 + 0.707i)53-s − i·55-s + ⋯
L(s)  = 1  + (0.422 − 0.906i)5-s + (0.906 − 0.422i)11-s + (0.0871 + 0.996i)13-s + (0.5 − 0.866i)17-s + (0.258 − 0.965i)19-s + (−0.984 − 0.173i)23-s + (−0.642 − 0.766i)25-s + (−0.996 − 0.0871i)29-s + (−0.173 + 0.984i)31-s + (−0.258 − 0.965i)37-s + (−0.642 + 0.766i)41-s + (−0.906 + 0.422i)43-s + (−0.173 − 0.984i)47-s + (−0.707 + 0.707i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.995 - 0.0943i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (5501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.995 - 0.0943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03913122480 - 0.8272186728i\)
\(L(\frac12)\) \(\approx\) \(0.03913122480 - 0.8272186728i\)
\(L(1)\) \(\approx\) \(0.9739847433 - 0.2854968181i\)
\(L(1)\) \(\approx\) \(0.9739847433 - 0.2854968181i\)

Euler product

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

−17.97463904050587300645545125583, −17.36933749070412769920331634969, −16.90034714911411936932993203752, −16.05045533825615650713566185529, −15.13358144068497065023630043180, −14.81239062129655394424903975931, −14.226541202509301609485008376728, −13.45509198701416170410652146794, −12.828248076482950742897550211506, −11.98558417478947545630778657299, −11.51204343975213220865020816074, −10.55887976472236693874234998432, −10.08447222963079840021207208957, −9.63670353012797754793098654569, −8.63904961258457350993704466078, −7.871164780956957444139036306479, −7.33538684243161256679137857688, −6.455689562724875499573505091703, −5.88939430728590598312863150136, −5.37513016395544793299641087066, −4.109897716000254733276605185661, −3.61736821374933315754876763293, −2.91258573817586756851071143695, −1.849166995716602366552118951349, −1.43055824043656296842122776603, 0.18890995851201774919654817300, 1.30770366289659602198658506699, 1.80148018557867055780692524344, 2.82926475797512825668176721149, 3.75653056464726698552562807447, 4.40714519728964588984336602820, 5.16627570602424356554764393485, 5.76288432199785109291925703210, 6.655986712000479254546348967494, 7.138723088305404939672134694527, 8.22913136838232818936377155022, 8.74197404786118233759191232023, 9.50135867615018242218472895024, 9.72580912010099205330766503517, 10.86559954873509526732302404032, 11.71566946139590907762851628733, 11.89191403684937366630210847938, 12.87742571382280940046354881971, 13.45562700201677598330678849675, 14.181330999319316060038660110112, 14.468655863426709565462840026914, 15.65297266905626677997997233626, 16.229191329571864297907622495593, 16.64432848428335152825297636490, 17.30150000793054217085468020949

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