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Properties

Label 1-6048-6048.5011-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.814 - 0.579i$
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  + (−0.996 + 0.0871i)5-s + (0.0871 − 0.996i)11-s + (−0.819 + 0.573i)13-s + (−0.5 + 0.866i)17-s + (0.258 − 0.965i)19-s + (0.342 + 0.939i)23-s + (0.984 − 0.173i)25-s + (0.573 − 0.819i)29-s + (−0.939 + 0.342i)31-s + (0.258 + 0.965i)37-s + (−0.984 − 0.173i)41-s + (0.0871 − 0.996i)43-s + (0.939 + 0.342i)47-s + (−0.707 + 0.707i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.996 + 0.0871i)5-s + (0.0871 − 0.996i)11-s + (−0.819 + 0.573i)13-s + (−0.5 + 0.866i)17-s + (0.258 − 0.965i)19-s + (0.342 + 0.939i)23-s + (0.984 − 0.173i)25-s + (0.573 − 0.819i)29-s + (−0.939 + 0.342i)31-s + (0.258 + 0.965i)37-s + (−0.984 − 0.173i)41-s + (0.0871 − 0.996i)43-s + (0.939 + 0.342i)47-s + (−0.707 + 0.707i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.814 - 0.579i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (5011, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.814 - 0.579i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09880254154 - 0.3094338866i\)
\(L(\frac12)\) \(\approx\) \(0.09880254154 - 0.3094338866i\)
\(L(1)\) \(\approx\) \(0.7499435697 + 0.02460478798i\)
\(L(1)\) \(\approx\) \(0.7499435697 + 0.02460478798i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.996 + 0.0871i)T \)
11 \( 1 + (0.0871 - 0.996i)T \)
13 \( 1 + (-0.819 + 0.573i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.258 - 0.965i)T \)
23 \( 1 + (0.342 + 0.939i)T \)
29 \( 1 + (0.573 - 0.819i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (0.258 + 0.965i)T \)
41 \( 1 + (-0.984 - 0.173i)T \)
43 \( 1 + (0.0871 - 0.996i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (-0.707 + 0.707i)T \)
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 \( 1 + (-0.573 + 0.819i)T \)
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

−18.01639782165212507408919677227, −17.34650593267001158797408456492, −16.45433100006794242852119583685, −16.10992311355719617786568358426, −15.316974519633547806334686062816, −14.584786429372825565064114353941, −14.444764980834720533793336751008, −13.10182239323206272776506343624, −12.69381284714707289413436126115, −12.013858228841407715630591212401, −11.55175483990476168215678365367, −10.61255100476038594532603502108, −10.115716139761682725421713964304, −9.25840747348100507294495961180, −8.62769228460507057043631483270, −7.77731781678039998051000193125, −7.28356196812167409164979060613, −6.77106463189573092491011912257, −5.669044946815367023228578039, −4.85455423043700997481477610705, −4.438814759386686088727214299243, −3.55188274780003059092503958833, −2.8003840346065630203179398494, −2.01053290272166511401042271360, −0.91493746676236749132467111005, 0.10365836981121473143318933477, 1.11659084497903728686449185735, 2.166929167777467555951966944713, 3.051933254086606730575834568413, 3.65996735758646517471702058323, 4.426828773338906052172859587, 5.07614474634194783474742783827, 5.9658057010561175531516048287, 6.81094178349360564722910479369, 7.29476734115565407530619243728, 8.114110078299902122908884945752, 8.71307507191808074241243402323, 9.31914258669920273235803636708, 10.230996418517126916359020007263, 11.04131427551171394990706929230, 11.43086200641534727046864799142, 12.07524199733926738615769158047, 12.78191333649724856487788960244, 13.591565346876019622325045902285, 14.114400694711629618769170166769, 15.00169553102920333233380246975, 15.49236183655775119523537127036, 15.978766253695644079208723350526, 16.99518029858354512466342434372, 17.13583153093430083703691659730

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