Properties

Label 1-6048-6048.3317-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: $\chi_{6048} (3317, \cdot )$
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

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

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