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Properties

Label 1-6037-6037.63-r1-0-0
Degree $1$
Conductor $6037$
Sign $0.457 - 0.889i$
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  + (0.0811 + 0.996i)2-s + (0.993 + 0.114i)3-s + (−0.986 + 0.161i)4-s + (0.379 + 0.925i)5-s + (−0.0333 + 0.999i)6-s + (−0.759 + 0.650i)7-s + (−0.241 − 0.970i)8-s + (0.973 + 0.227i)9-s + (−0.891 + 0.453i)10-s + (−0.697 − 0.716i)11-s + (−0.998 + 0.0478i)12-s + (0.673 − 0.739i)13-s + (−0.710 − 0.703i)14-s + (0.271 + 0.962i)15-s + (0.947 − 0.319i)16-s + (−0.770 + 0.637i)17-s + ⋯
L(s)  = 1  + (0.0811 + 0.996i)2-s + (0.993 + 0.114i)3-s + (−0.986 + 0.161i)4-s + (0.379 + 0.925i)5-s + (−0.0333 + 0.999i)6-s + (−0.759 + 0.650i)7-s + (−0.241 − 0.970i)8-s + (0.973 + 0.227i)9-s + (−0.891 + 0.453i)10-s + (−0.697 − 0.716i)11-s + (−0.998 + 0.0478i)12-s + (0.673 − 0.739i)13-s + (−0.710 − 0.703i)14-s + (0.271 + 0.962i)15-s + (0.947 − 0.319i)16-s + (−0.770 + 0.637i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6037\)
Sign: $0.457 - 0.889i$
Analytic conductor: \(648.765\)
Root analytic conductor: \(648.765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6037} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6037,\ (1:\ ),\ 0.457 - 0.889i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6119530923 + 0.3733483621i\)
\(L(\frac12)\) \(\approx\) \(-0.6119530923 + 0.3733483621i\)
\(L(1)\) \(\approx\) \(0.8371812292 + 0.8880548435i\)
\(L(1)\) \(\approx\) \(0.8371812292 + 0.8880548435i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad6037 \( 1 \)
good2 \( 1 + (0.0811 + 0.996i)T \)
3 \( 1 + (0.993 + 0.114i)T \)
5 \( 1 + (0.379 + 0.925i)T \)
7 \( 1 + (-0.759 + 0.650i)T \)
11 \( 1 + (-0.697 - 0.716i)T \)
13 \( 1 + (0.673 - 0.739i)T \)
17 \( 1 + (-0.770 + 0.637i)T \)
19 \( 1 + (0.124 + 0.992i)T \)
23 \( 1 + (0.196 + 0.980i)T \)
29 \( 1 + (0.694 - 0.719i)T \)
31 \( 1 + (-0.650 - 0.759i)T \)
37 \( 1 + (0.981 + 0.191i)T \)
41 \( 1 + (0.672 + 0.739i)T \)
43 \( 1 + (0.539 - 0.842i)T \)
47 \( 1 + (0.0260 + 0.999i)T \)
53 \( 1 + (-0.293 - 0.956i)T \)
59 \( 1 + (0.861 + 0.508i)T \)
61 \( 1 + (-0.895 - 0.444i)T \)
67 \( 1 + (-0.892 + 0.450i)T \)
71 \( 1 + (0.540 + 0.841i)T \)
73 \( 1 + (-0.970 - 0.240i)T \)
79 \( 1 + (-0.932 + 0.361i)T \)
83 \( 1 + (-0.457 + 0.888i)T \)
89 \( 1 + (-0.546 + 0.837i)T \)
97 \( 1 + (0.557 - 0.830i)T \)
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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

−17.139941960685724328550034877049, −16.12055134505690950271825638484, −15.90596828880705777316380953219, −14.80021746443277559592712452512, −14.041039082840002126141888689024, −13.575248588477346193362340889811, −12.97528295010122276107568443172, −12.720151497651314692264484485903, −11.90345928219387832560160125191, −10.84548698631569458894146431635, −10.35208750958812752276115280842, −9.56774679295601792006524271962, −8.989972527780540058743525782984, −8.79831291651351876306813390415, −7.72596466281408577002103862146, −7.00120913846662231931177290485, −6.18410732081765285931991496215, −4.97198515030315922067077191057, −4.49031270238245140500269386048, −3.97359168115066973334519210707, −2.9106113272205946765633574049, −2.49249118848017183121720360158, −1.61147122420194762990984845386, −0.90356876258711453322854650750, −0.084845575012955202998426261231, 1.22946643945105239423118910463, 2.41283688637755263804991045640, 3.01873146848728270349950576065, 3.61275781890426386991851393853, 4.29136566716838042559427248939, 5.7037829216984692220786596908, 5.79829542078552934432616312192, 6.59693833712865179821131301371, 7.4396023008303568285193405772, 8.060096062981137301330899725269, 8.54495901063396807160579911138, 9.40587409542144045607192183548, 9.86825418168622898443453421356, 10.53738497398744693965707639063, 11.378841850785499241358964463772, 12.68941732789798655746518947795, 13.07931305980126011548814209564, 13.61145181443259301901255007708, 14.20321500858495416566159092840, 15.04989069520310281795612409479, 15.30896999966321080370499092129, 15.93804326396150031661384234773, 16.46696167559680730093555529032, 17.58259672237764051548584204358, 18.09043043429056739194733220335

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