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Properties

Label 1-6027-6027.65-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.0957 - 0.995i$
Analytic cond. $27.9892$
Root an. cond. $27.9892$
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  + (−0.486 + 0.873i)2-s + (−0.525 − 0.850i)4-s + (−0.967 + 0.251i)5-s + (0.998 − 0.0448i)8-s + (0.251 − 0.967i)10-s + (−0.989 − 0.141i)11-s + (0.244 − 0.969i)13-s + (−0.447 + 0.894i)16-s + (0.685 + 0.727i)17-s + (−0.838 − 0.544i)19-s + (0.722 + 0.691i)20-s + (0.605 − 0.795i)22-s + (−0.280 + 0.959i)23-s + (0.873 − 0.486i)25-s + (0.727 + 0.685i)26-s + ⋯
L(s)  = 1  + (−0.486 + 0.873i)2-s + (−0.525 − 0.850i)4-s + (−0.967 + 0.251i)5-s + (0.998 − 0.0448i)8-s + (0.251 − 0.967i)10-s + (−0.989 − 0.141i)11-s + (0.244 − 0.969i)13-s + (−0.447 + 0.894i)16-s + (0.685 + 0.727i)17-s + (−0.838 − 0.544i)19-s + (0.722 + 0.691i)20-s + (0.605 − 0.795i)22-s + (−0.280 + 0.959i)23-s + (0.873 − 0.486i)25-s + (0.727 + 0.685i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0957 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0957 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.0957 - 0.995i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.0957 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06470779760 - 0.07123117040i\)
\(L(\frac12)\) \(\approx\) \(0.06470779760 - 0.07123117040i\)
\(L(1)\) \(\approx\) \(0.5130151569 + 0.2064517176i\)
\(L(1)\) \(\approx\) \(0.5130151569 + 0.2064517176i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.486 + 0.873i)T \)
5 \( 1 + (-0.967 + 0.251i)T \)
11 \( 1 + (-0.989 - 0.141i)T \)
13 \( 1 + (0.244 - 0.969i)T \)
17 \( 1 + (0.685 + 0.727i)T \)
19 \( 1 + (-0.838 - 0.544i)T \)
23 \( 1 + (-0.280 + 0.959i)T \)
29 \( 1 + (-0.569 + 0.822i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (-0.646 - 0.762i)T \)
43 \( 1 + (-0.351 + 0.936i)T \)
47 \( 1 + (0.273 + 0.961i)T \)
53 \( 1 + (0.973 + 0.229i)T \)
59 \( 1 + (0.772 - 0.635i)T \)
61 \( 1 + (-0.237 + 0.971i)T \)
67 \( 1 + (0.629 - 0.777i)T \)
71 \( 1 + (-0.822 + 0.569i)T \)
73 \( 1 + (0.680 - 0.733i)T \)
79 \( 1 + (0.965 + 0.258i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (0.696 + 0.717i)T \)
97 \( 1 + (0.987 + 0.156i)T \)
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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

−18.28645358116712328899198804423, −17.1154677182939815873493336143, −16.63659279442793789585847434046, −16.16393275375891269184890445909, −15.35439498736461397670916336865, −14.59047166020724268494096418425, −13.76540698016263624953566797215, −13.0834220372379607849211426671, −12.46192134767973841903857475776, −11.8190467412473095519821654804, −11.423122197759661324294204692, −10.51698884365179398257788961414, −10.13330802341456342866720573145, −9.17158818098877827889124682831, −8.59081376556055682019087182864, −7.98628892552922224742247929719, −7.39114241170990962806655170188, −6.66307948198055089778168811487, −5.41663607425741695575001257440, −4.75371782220639316484033465364, −3.94106048094884642839085751633, −3.55422002193486941038517532158, −2.4544612176084737456270739527, −1.94711819479841863082165594746, −0.7810263643630129582755290977, 0.04357565419912844233587151658, 1.0371696754329612673872824610, 2.08804159813662745212766609482, 3.231563629880076572766462032797, 3.79097715436494307048939924405, 4.766092609777392946858423082336, 5.44606938585391770263115068999, 6.00903827965068121595092204999, 6.98321469225888385290063063590, 7.55808249659960719395349182045, 8.04450835099937170326536829311, 8.625191456696477038710408630825, 9.38484922404606348387989387333, 10.454722337507387384910090808591, 10.637545510931691491352274788332, 11.34354732698686454117318224518, 12.49999507914438853597861783409, 12.92721202282330289857454675804, 13.68745366120595397709520496326, 14.68445801561331251182164821650, 14.96139450694847384962320116938, 15.66919581343776070656550900149, 16.11490487682517822350325377921, 16.751348072746448312330804123149, 17.65383515741692012607595533860

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