Properties

Label 1-837-837.149-r1-0-0
Degree $1$
Conductor $837$
Sign $0.443 - 0.896i$
Analytic cond. $89.9481$
Root an. cond. $89.9481$
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  + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (−0.766 + 0.642i)5-s + (−0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.939 − 0.342i)11-s + (0.173 + 0.984i)13-s + (−0.766 + 0.642i)14-s + (0.173 − 0.984i)16-s − 17-s + (−0.5 − 0.866i)19-s + (−0.173 + 0.984i)20-s + (0.766 − 0.642i)22-s + (−0.766 + 0.642i)23-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (−0.766 + 0.642i)5-s + (−0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.939 − 0.342i)11-s + (0.173 + 0.984i)13-s + (−0.766 + 0.642i)14-s + (0.173 − 0.984i)16-s − 17-s + (−0.5 − 0.866i)19-s + (−0.173 + 0.984i)20-s + (0.766 − 0.642i)22-s + (−0.766 + 0.642i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.443 - 0.896i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.443 - 0.896i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.293208608 - 1.424175333i\)
\(L(\frac12)\) \(\approx\) \(2.293208608 - 1.424175333i\)
\(L(1)\) \(\approx\) \(1.496450412 - 0.2985754547i\)
\(L(1)\) \(\approx\) \(1.496450412 - 0.2985754547i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 + (0.766 - 0.642i)T \)
7 \( 1 + (0.939 - 0.342i)T \)
11 \( 1 + (-0.939 + 0.342i)T \)
13 \( 1 + (-0.173 - 0.984i)T \)
17 \( 1 + T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 - T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (0.766 + 0.642i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (0.766 + 0.642i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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

−22.48311035795531282973343406384, −21.41017566028574226979507563046, −20.219886427377468187056555679774, −20.08553140036482285702118945239, −19.26300337956083018156467829901, −17.863335530961581217743160839966, −16.94183726025533924311541905769, −16.28746211552490575255578615043, −15.671707126664982289514260299829, −14.873183695887769456680395552704, −14.00327828356577632208218419999, −12.87940633143001428287564948584, −12.65161790101693381674471329871, −11.740416828606581903953494276506, −10.81803354299280156498167165862, −9.75572856569318413640346401969, −8.5350627530195505720368725678, −7.89434171127679073988262313358, −6.75081491217157186393780962474, −6.231743080351943203394972124206, −5.02019570017142467573223500790, −4.079158555648139256677513795935, −3.60845209662999696935315956841, −2.37416598637557456955414361623, −0.88295283714498460784705887495, 0.51929276631666310471704851107, 2.08367570763085195955824169562, 2.941232798742905962359147149516, 3.93054659564947608939305961970, 4.40941527979310314869174524820, 5.94959300265214735580208642245, 6.589024798049955282412154450221, 7.147985237385233072452520826410, 8.639454460094191530443167260870, 9.53864017508717344689076364102, 10.55240582218375538814153506684, 11.541671577438191329148375953416, 11.78601252669882132649273874069, 12.88644306209459666816276356125, 13.70832192701210636226538330114, 14.43775882039540070572045218298, 15.36573296333511264534124195681, 15.873394095032890019070444986491, 16.664332278090784079488915221355, 17.991665628121624689477022497994, 19.06892824011827060837639207998, 19.52381633856871348103376680132, 19.95481965618330468435205358260, 21.32027287782387744337290205213, 22.002249054480504323940476037214

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