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Properties

Label 1-6047-6047.2-r0-0-0
Degree $1$
Conductor $6047$
Sign $0.992 + 0.125i$
Analytic cond. $28.0821$
Root an. cond. $28.0821$
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  + (0.437 + 0.899i)2-s + (−0.648 + 0.761i)3-s + (−0.617 + 0.786i)4-s + (−0.212 + 0.977i)5-s + (−0.968 − 0.249i)6-s + (0.989 + 0.147i)7-s + (−0.977 − 0.211i)8-s + (−0.159 − 0.987i)9-s + (−0.971 + 0.235i)10-s + (−0.356 + 0.934i)11-s + (−0.198 − 0.980i)12-s + (−0.633 + 0.773i)13-s + (0.300 + 0.953i)14-s + (−0.606 − 0.795i)15-s + (−0.237 − 0.971i)16-s + (−0.950 − 0.309i)17-s + ⋯
L(s)  = 1  + (0.437 + 0.899i)2-s + (−0.648 + 0.761i)3-s + (−0.617 + 0.786i)4-s + (−0.212 + 0.977i)5-s + (−0.968 − 0.249i)6-s + (0.989 + 0.147i)7-s + (−0.977 − 0.211i)8-s + (−0.159 − 0.987i)9-s + (−0.971 + 0.235i)10-s + (−0.356 + 0.934i)11-s + (−0.198 − 0.980i)12-s + (−0.633 + 0.773i)13-s + (0.300 + 0.953i)14-s + (−0.606 − 0.795i)15-s + (−0.237 − 0.971i)16-s + (−0.950 − 0.309i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6047\)
Sign: $0.992 + 0.125i$
Analytic conductor: \(28.0821\)
Root analytic conductor: \(28.0821\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6047} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6047,\ (0:\ ),\ 0.992 + 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4599097316 + 0.02902942562i\)
\(L(\frac12)\) \(\approx\) \(-0.4599097316 + 0.02902942562i\)
\(L(1)\) \(\approx\) \(0.2852772327 + 0.7253937750i\)
\(L(1)\) \(\approx\) \(0.2852772327 + 0.7253937750i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad6047 \( 1 \)
good2 \( 1 + (0.437 + 0.899i)T \)
3 \( 1 + (-0.648 + 0.761i)T \)
5 \( 1 + (-0.212 + 0.977i)T \)
7 \( 1 + (0.989 + 0.147i)T \)
11 \( 1 + (-0.356 + 0.934i)T \)
13 \( 1 + (-0.633 + 0.773i)T \)
17 \( 1 + (-0.950 - 0.309i)T \)
19 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (-0.657 + 0.753i)T \)
29 \( 1 + (-0.890 - 0.455i)T \)
31 \( 1 + (0.468 + 0.883i)T \)
37 \( 1 + (0.302 - 0.953i)T \)
41 \( 1 + (0.119 + 0.992i)T \)
43 \( 1 + (-0.885 + 0.464i)T \)
47 \( 1 + (0.895 - 0.445i)T \)
53 \( 1 + (-0.893 + 0.449i)T \)
59 \( 1 + (-0.995 + 0.0944i)T \)
61 \( 1 + (-0.999 - 0.00935i)T \)
67 \( 1 + (-0.654 + 0.756i)T \)
71 \( 1 + (0.745 - 0.666i)T \)
73 \( 1 + (0.835 + 0.549i)T \)
79 \( 1 + (-0.00675 - 0.999i)T \)
83 \( 1 + (-0.704 - 0.709i)T \)
89 \( 1 + (-0.720 + 0.693i)T \)
97 \( 1 + (-0.425 + 0.905i)T \)
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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

−17.105387106730497532256283470103, −16.724292324662970795112981188019, −15.46272727114944250183983576096, −15.19142973307378644469436132852, −13.866138628118032924789579384712, −13.747587442019966488716586679532, −12.83376516499272914534802404726, −12.539610258356570597991977999732, −11.78706104259631659787113344701, −11.04507181180162711648942087859, −10.90927883650894530895552728939, −9.93519052759777070918541610774, −8.92105691396230763666666622914, −8.32485533227871533465213283403, −7.86268149302578258172619574280, −6.80024562859825836364334598389, −5.83481562737788096905888975153, −5.42796233560513899197178087466, −4.60964638280872024775979280609, −4.33244120056820108197762051782, −3.051179274606455147020197748315, −2.193177357960960605039304318598, −1.637260103519869430789190428210, −0.65640268217063296422966043698, −0.15392196203261881451704198243, 1.76764490935218589994821392073, 2.594341514436940546643807716067, 3.60341858211872148875435921258, 4.36425341473487048638431178098, 4.64610118949018689361614883381, 5.515307227686577590483924011605, 6.19488639057121546718037840314, 6.86354611114738082289533995150, 7.53980941729157588104517344122, 8.08399749911248169796018316718, 9.159997580444625854180106135555, 9.649343613265486872034510120404, 10.49729160845871296151490411567, 11.16294082613325054543850215977, 11.92199816637567307146120974759, 12.18725369736189996414302118931, 13.286276440097202948670430902225, 14.2089987077304574304023486481, 14.54068627029174897902556039184, 15.226897979920864039964061008665, 15.52170140825231121498745220840, 16.3109636770846769797069320970, 17.10298286529978225297437775009, 17.54170363864024271495979526891, 18.20383697232246042720207123546

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