↑

Properties

Label 1-6037-6037.23-r1-0-0
Degree $1$
Conductor $6037$
Sign $0.829 + 0.557i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.229 − 0.973i)2-s + (−0.746 − 0.665i)3-s + (−0.895 + 0.445i)4-s + (0.616 − 0.787i)5-s + (−0.477 + 0.878i)6-s + (0.996 − 0.0842i)7-s + (0.639 + 0.769i)8-s + (0.113 + 0.993i)9-s + (−0.907 − 0.419i)10-s + (−0.811 + 0.584i)11-s + (0.964 + 0.263i)12-s + (0.151 + 0.988i)13-s + (−0.310 − 0.950i)14-s + (−0.984 + 0.177i)15-s + (0.602 − 0.798i)16-s + (0.993 − 0.114i)17-s + ⋯
L(s)  = 1  + (−0.229 − 0.973i)2-s + (−0.746 − 0.665i)3-s + (−0.895 + 0.445i)4-s + (0.616 − 0.787i)5-s + (−0.477 + 0.878i)6-s + (0.996 − 0.0842i)7-s + (0.639 + 0.769i)8-s + (0.113 + 0.993i)9-s + (−0.907 − 0.419i)10-s + (−0.811 + 0.584i)11-s + (0.964 + 0.263i)12-s + (0.151 + 0.988i)13-s + (−0.310 − 0.950i)14-s + (−0.984 + 0.177i)15-s + (0.602 − 0.798i)16-s + (0.993 − 0.114i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.829 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.829 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6037\)
Sign: $0.829 + 0.557i$
Analytic conductor: \(648.765\)
Root analytic conductor: \(648.765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6037} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6037,\ (1:\ ),\ 0.829 + 0.557i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8504971888 + 0.2593364885i\)
\(L(\frac12)\) \(\approx\) \(0.8504971888 + 0.2593364885i\)
\(L(1)\) \(\approx\) \(0.6818855508 - 0.4321047116i\)
\(L(1)\) \(\approx\) \(0.6818855508 - 0.4321047116i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad6037 \( 1 \)
good2 \( 1 + (-0.229 - 0.973i)T \)
3 \( 1 + (-0.746 - 0.665i)T \)
5 \( 1 + (0.616 - 0.787i)T \)
7 \( 1 + (0.996 - 0.0842i)T \)
11 \( 1 + (-0.811 + 0.584i)T \)
13 \( 1 + (0.151 + 0.988i)T \)
17 \( 1 + (0.993 - 0.114i)T \)
19 \( 1 + (0.882 + 0.470i)T \)
23 \( 1 + (-0.391 - 0.920i)T \)
29 \( 1 + (-0.168 + 0.985i)T \)
31 \( 1 + (-0.934 + 0.355i)T \)
37 \( 1 + (-0.974 - 0.223i)T \)
41 \( 1 + (-0.991 + 0.126i)T \)
43 \( 1 + (0.230 + 0.973i)T \)
47 \( 1 + (0.770 + 0.637i)T \)
53 \( 1 + (-0.948 + 0.318i)T \)
59 \( 1 + (-0.107 + 0.994i)T \)
61 \( 1 + (0.874 + 0.485i)T \)
67 \( 1 + (-0.392 - 0.919i)T \)
71 \( 1 + (-0.710 + 0.703i)T \)
73 \( 1 + (0.826 - 0.563i)T \)
79 \( 1 + (0.997 + 0.0759i)T \)
83 \( 1 + (0.254 + 0.967i)T \)
89 \( 1 + (0.921 + 0.388i)T \)
97 \( 1 + (0.981 + 0.191i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.40366184268646643791254241058, −17.0607110705913374739737663372, −15.99330372979408822173600666281, −15.61539055850468974946311807884, −15.00833767028019908784001915908, −14.42756614165871391121657438106, −13.70012326619011828487229651769, −13.19606217644858169548218701368, −12.07929229896725705476439805530, −11.34964710690804358024291507549, −10.60573790172800190745714331715, −10.248298099408580171032490478342, −9.57400434402801311558014401806, −8.80960136553434758875189158658, −7.84829263393011412632315700176, −7.52547012533956906686398702139, −6.572536647825034318224180382475, −5.715160066785997817415901536437, −5.39182831691408411192092382022, −5.05811342456426801071926228042, −3.73236042986071119613911515670, −3.310154402497525119274924861395, −1.9951855864061405740446607166, −0.98035261899588118760575274946, −0.17762659511565608748428238206, 0.895141542959528690951892972037, 1.55445880719236907675882673443, 1.874392909305026156320323523152, 2.79442395156449700767194872427, 3.99875252597760587990692152287, 4.827105541105688180481417350773, 5.1682542080842949952108841076, 5.80355053220828667721453186625, 6.98310307957520202901502031830, 7.71881479589119174099511344354, 8.22606850245814129025850606304, 9.03280377994024604259519211935, 9.74935006745989455732754949194, 10.54223880778151498905820786008, 10.89956078876337864871415936343, 11.984647214006973252738591989086, 12.12546779684077558503344000459, 12.75666711474080843525333303747, 13.54583078604074941925244679860, 14.075178120148840208779718769838, 14.590095998835546619295034153301, 16.11920478621836374801251290000, 16.49367280439611237469724210199, 17.134954338182250303769029900187, 17.80091437030509154914104019212

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