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Properties

Label 1-6048-6048.787-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.986 + 0.162i$
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  + (−0.0871 + 0.996i)5-s + (0.996 − 0.0871i)11-s + (−0.996 − 0.0871i)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 + (−0.0871 − 0.996i)29-s + (0.766 − 0.642i)31-s + (0.707 − 0.707i)37-s + (−0.642 − 0.766i)41-s + (−0.422 + 0.906i)43-s + (−0.766 − 0.642i)47-s + (0.965 + 0.258i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.0871 + 0.996i)5-s + (0.996 − 0.0871i)11-s + (−0.996 − 0.0871i)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 + (−0.0871 − 0.996i)29-s + (0.766 − 0.642i)31-s + (0.707 − 0.707i)37-s + (−0.642 − 0.766i)41-s + (−0.422 + 0.906i)43-s + (−0.766 − 0.642i)47-s + (0.965 + 0.258i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.986 + 0.162i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (787, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.986 + 0.162i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.566568724 + 0.1280466507i\)
\(L(\frac12)\) \(\approx\) \(1.566568724 + 0.1280466507i\)
\(L(1)\) \(\approx\) \(1.026342047 + 0.1332329721i\)
\(L(1)\) \(\approx\) \(1.026342047 + 0.1332329721i\)

Euler product

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

−17.45167826991526388828600359808, −17.07771760508099910603711621021, −16.26364327711392163165384738755, −16.05649509044060878086674412151, −14.93092374801427364691127385606, −14.454124300518220069457356603865, −13.79542893581264621707249713707, −12.99173665891250526800121920544, −12.34468864191189811551344415545, −11.85564321271465846386095567799, −11.33648411275595325880755380685, −10.06381688096366406629226175243, −9.836313874702313301016210835810, −8.91585711469629915034606183577, −8.48162259932657907025115497639, −7.67944062252987027927406964584, −6.85960656894475095379230736805, −6.296311633325663753454887893991, −5.23247802037081591577216869760, −4.801651434947355144247469766325, −4.11252129893435764220360338937, −3.27770298954696405238712202593, −2.30044561421181365918166848966, −1.50405594638515314661830179739, −0.701882797697702885940127402235, 0.55598593066251879612892147221, 1.8281791925417792460619224340, 2.41300217514247026271060713058, 3.26106233925326626624238445476, 3.98927279882365109032081033487, 4.619301244359805892931510879251, 5.69828088781644423410213503427, 6.28219343939504807877302513225, 6.98382439450778179769426767640, 7.52074156208215804239373915509, 8.30051183169394020023667261487, 9.187210803992305540566279595, 9.81607939314934887627985425401, 10.34861052908569218972279065675, 11.3516384982925907241436435278, 11.56740110552244815820542543922, 12.32719035841303900467949874577, 13.35134730897834455292447989347, 13.72855033931837869078897445736, 14.60394130132627339641528816631, 15.112026325511399916210091295085, 15.467301049339410866951875390293, 16.52492034791728573350093415595, 17.19168070815895411466353439765, 17.685791623632496189600265685352

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