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Properties

Label 1-6037-6037.66-r1-0-0
Degree $1$
Conductor $6037$
Sign $0.855 - 0.518i$
Analytic cond. $648.765$
Root an. cond. $648.765$
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  + (0.342 + 0.939i)2-s + (0.741 + 0.670i)3-s + (−0.765 + 0.643i)4-s + (0.0145 − 0.999i)5-s + (−0.375 + 0.926i)6-s + (−0.594 − 0.803i)7-s + (−0.867 − 0.498i)8-s + (0.100 + 0.994i)9-s + (0.944 − 0.328i)10-s + (0.983 − 0.180i)11-s + (−0.999 − 0.0353i)12-s + (−0.631 − 0.775i)13-s + (0.551 − 0.834i)14-s + (0.681 − 0.732i)15-s + (0.170 − 0.985i)16-s + (0.797 + 0.603i)17-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)2-s + (0.741 + 0.670i)3-s + (−0.765 + 0.643i)4-s + (0.0145 − 0.999i)5-s + (−0.375 + 0.926i)6-s + (−0.594 − 0.803i)7-s + (−0.867 − 0.498i)8-s + (0.100 + 0.994i)9-s + (0.944 − 0.328i)10-s + (0.983 − 0.180i)11-s + (−0.999 − 0.0353i)12-s + (−0.631 − 0.775i)13-s + (0.551 − 0.834i)14-s + (0.681 − 0.732i)15-s + (0.170 − 0.985i)16-s + (0.797 + 0.603i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6037\)
Sign: $0.855 - 0.518i$
Analytic conductor: \(648.765\)
Root analytic conductor: \(648.765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6037} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6037,\ (1:\ ),\ 0.855 - 0.518i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.153708705 - 0.8815417166i\)
\(L(\frac12)\) \(\approx\) \(3.153708705 - 0.8815417166i\)
\(L(1)\) \(\approx\) \(1.427298982 + 0.4377220141i\)
\(L(1)\) \(\approx\) \(1.427298982 + 0.4377220141i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad6037 \( 1 \)
good2 \( 1 + (0.342 + 0.939i)T \)
3 \( 1 + (0.741 + 0.670i)T \)
5 \( 1 + (0.0145 - 0.999i)T \)
7 \( 1 + (-0.594 - 0.803i)T \)
11 \( 1 + (0.983 - 0.180i)T \)
13 \( 1 + (-0.631 - 0.775i)T \)
17 \( 1 + (0.797 + 0.603i)T \)
19 \( 1 + (0.790 - 0.612i)T \)
23 \( 1 + (-0.126 - 0.991i)T \)
29 \( 1 + (0.501 - 0.864i)T \)
31 \( 1 + (0.898 + 0.439i)T \)
37 \( 1 + (-0.581 - 0.813i)T \)
41 \( 1 + (0.979 + 0.202i)T \)
43 \( 1 + (0.388 - 0.921i)T \)
47 \( 1 + (0.827 - 0.560i)T \)
53 \( 1 + (-0.958 - 0.284i)T \)
59 \( 1 + (0.992 - 0.120i)T \)
61 \( 1 + (0.997 - 0.0676i)T \)
67 \( 1 + (0.979 - 0.199i)T \)
71 \( 1 + (0.530 - 0.847i)T \)
73 \( 1 + (-0.999 + 0.0426i)T \)
79 \( 1 + (-0.334 + 0.942i)T \)
83 \( 1 + (-0.469 - 0.883i)T \)
89 \( 1 + (-0.718 + 0.695i)T \)
97 \( 1 + (0.973 - 0.230i)T \)
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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

−17.83699574496782000047482048997, −17.34066353533673139984956754130, −16.07180019867328279695629595547, −15.38191065339166126713201274269, −14.57000887235154230242850769697, −14.22503169597507548230275377610, −13.86049770527423135421584004892, −12.89327990727532525662284899163, −12.22507053734741940933808299803, −11.775916437568259722473312344666, −11.36110773640431914811794275782, −10.01610997758654044689382118248, −9.69673048174773484562975074441, −9.234062865197313699058465872857, −8.36692836733479145012855543623, −7.44465152703948017899530121043, −6.802727894661225088385347131133, −6.11559649097538765374886891530, −5.45397503360182771007674700218, −4.28621882912682561663505714062, −3.54385617264708793314233417758, −2.973193264431829663049848081674, −2.48614482788591742396780755246, −1.63995849803416642613797592786, −0.97030516484898900143137205797, 0.411704711311194953392765936716, 0.98960835050712825697745914685, 2.43782220268703549986503414526, 3.31776560801648781535298896951, 3.94862405828378318680679794070, 4.42837430417861895013802919860, 5.19940379526351913869939551739, 5.83405612781886138767540926599, 6.77523359391437264274513742021, 7.48434293537427815308637873041, 8.172196158684589407104351608929, 8.65382082586995336849340607608, 9.48263471017081399897554351904, 9.83010633584521470508041657512, 10.58993316418484654244043359650, 11.815935834604301851973771655397, 12.525332882982406978663197098, 13.00308670930928373357015794157, 13.87505130468434756119672000603, 14.12283634943898066000940965034, 14.86272509936065416461558870933, 15.74559797144839150149514136266, 16.001480056654394559437416236285, 16.74784883919980194738270661565, 17.214350613231444036833963487520

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