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Properties

Label 1-6008-6008.477-r0-0-0
Degree $1$
Conductor $6008$
Sign $-0.0996 + 0.995i$
Analytic cond. $27.9010$
Root an. cond. $27.9010$
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  + (−0.868 − 0.496i)3-s + (−0.549 + 0.835i)5-s + (0.402 − 0.915i)7-s + (0.507 + 0.861i)9-s + (−0.783 − 0.621i)11-s + (−0.999 + 0.0335i)13-s + (0.891 − 0.452i)15-s + (0.324 + 0.945i)17-s + (−0.563 − 0.825i)19-s + (−0.804 + 0.594i)21-s + (0.991 − 0.133i)23-s + (−0.395 − 0.918i)25-s + (−0.0125 − 0.999i)27-s + (0.316 + 0.948i)29-s + (−0.916 − 0.399i)31-s + ⋯
L(s)  = 1  + (−0.868 − 0.496i)3-s + (−0.549 + 0.835i)5-s + (0.402 − 0.915i)7-s + (0.507 + 0.861i)9-s + (−0.783 − 0.621i)11-s + (−0.999 + 0.0335i)13-s + (0.891 − 0.452i)15-s + (0.324 + 0.945i)17-s + (−0.563 − 0.825i)19-s + (−0.804 + 0.594i)21-s + (0.991 − 0.133i)23-s + (−0.395 − 0.918i)25-s + (−0.0125 − 0.999i)27-s + (0.316 + 0.948i)29-s + (−0.916 − 0.399i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0996 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0996 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $-0.0996 + 0.995i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (477, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ -0.0996 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1953718227 + 0.2159105797i\)
\(L(\frac12)\) \(\approx\) \(0.1953718227 + 0.2159105797i\)
\(L(1)\) \(\approx\) \(0.6113352769 - 0.07674717326i\)
\(L(1)\) \(\approx\) \(0.6113352769 - 0.07674717326i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (-0.868 - 0.496i)T \)
5 \( 1 + (-0.549 + 0.835i)T \)
7 \( 1 + (0.402 - 0.915i)T \)
11 \( 1 + (-0.783 - 0.621i)T \)
13 \( 1 + (-0.999 + 0.0335i)T \)
17 \( 1 + (0.324 + 0.945i)T \)
19 \( 1 + (-0.563 - 0.825i)T \)
23 \( 1 + (0.991 - 0.133i)T \)
29 \( 1 + (0.316 + 0.948i)T \)
31 \( 1 + (-0.916 - 0.399i)T \)
37 \( 1 + (0.995 - 0.0920i)T \)
41 \( 1 + (-0.929 + 0.368i)T \)
43 \( 1 + (0.888 - 0.459i)T \)
47 \( 1 + (-0.935 + 0.352i)T \)
53 \( 1 + (-0.968 + 0.248i)T \)
59 \( 1 + (0.923 + 0.383i)T \)
61 \( 1 + (0.268 - 0.963i)T \)
67 \( 1 + (-0.813 - 0.580i)T \)
71 \( 1 + (0.260 + 0.965i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.563 - 0.825i)T \)
83 \( 1 + (0.756 + 0.653i)T \)
89 \( 1 + (0.842 - 0.539i)T \)
97 \( 1 + (-0.994 - 0.108i)T \)
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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

−17.58429899604515604535646222098, −16.75249106381672951608311238769, −16.27078820932930442363575586800, −15.686882963835724969946093333751, −14.886957605181644978847637118831, −14.76791881363661984882261587947, −13.36844020902985297714798013268, −12.580600123264699867028818345850, −12.321439929507814311153208082712, −11.632552415502859885141825844549, −11.12144492897041567906873136746, −10.16090324468287168255512004956, −9.59021003552541521761407196912, −9.01853758454895137434224265409, −8.10225379189736649703106643365, −7.54645284527586597518784481556, −6.734429885040012166633998891626, −5.69768124781421259057565074921, −5.17091149629221655497991195861, −4.79835852313315311866002594369, −4.08561908819869637159790511522, −3.042821143549270720039629508318, −2.17964538013609162675639699189, −1.21411530035138615556179557976, −0.12173039738968492275214153207, 0.71739151547037898492690111428, 1.75016328105766781410366343535, 2.63725537207663488051053717348, 3.41036540415079861571477628441, 4.37484219747534993771148838909, 4.92936131882117077644174200851, 5.71676319249444053956806150494, 6.67435216890331206720124045535, 6.98071042037135935119744441843, 7.835821653124917268105068155083, 8.078203041522192152850439024051, 9.319468571617495031831862828916, 10.44568768808611360533385614397, 10.6173538766309316902609317935, 11.18625962908110510921086621252, 11.79925838916859604432773822979, 12.78642700675108025406060334021, 13.05294990971206810794501401185, 13.96766519573736574425965021285, 14.6828046483290690802254447520, 15.157691608081853115448645659804, 16.12188343535702917873317562533, 16.66089283234329777934883581362, 17.27636297212519421864767089614, 17.84988330505995466572899344215

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