Properties

Label 1-837-837.545-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.979 + 0.203i$
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  + (0.882 + 0.469i)2-s + (0.559 + 0.829i)4-s + (−0.173 − 0.984i)5-s + (−0.241 − 0.970i)7-s + (0.104 + 0.994i)8-s + (0.309 − 0.951i)10-s + (−0.961 − 0.275i)11-s + (0.0348 − 0.999i)13-s + (0.241 − 0.970i)14-s + (−0.374 + 0.927i)16-s + (0.104 + 0.994i)17-s + (0.309 − 0.951i)19-s + (0.719 − 0.694i)20-s + (−0.719 − 0.694i)22-s + (−0.961 + 0.275i)23-s + ⋯
L(s)  = 1  + (0.882 + 0.469i)2-s + (0.559 + 0.829i)4-s + (−0.173 − 0.984i)5-s + (−0.241 − 0.970i)7-s + (0.104 + 0.994i)8-s + (0.309 − 0.951i)10-s + (−0.961 − 0.275i)11-s + (0.0348 − 0.999i)13-s + (0.241 − 0.970i)14-s + (−0.374 + 0.927i)16-s + (0.104 + 0.994i)17-s + (0.309 − 0.951i)19-s + (0.719 − 0.694i)20-s + (−0.719 − 0.694i)22-s + (−0.961 + 0.275i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.979 + 0.203i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.979 + 0.203i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01780904024 + 0.1735383856i\)
\(L(\frac12)\) \(\approx\) \(0.01780904024 + 0.1735383856i\)
\(L(1)\) \(\approx\) \(1.288195520 + 0.07326372999i\)
\(L(1)\) \(\approx\) \(1.288195520 + 0.07326372999i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.882 + 0.469i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (-0.241 - 0.970i)T \)
11 \( 1 + (-0.961 - 0.275i)T \)
13 \( 1 + (0.0348 - 0.999i)T \)
17 \( 1 + (0.104 + 0.994i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.961 + 0.275i)T \)
29 \( 1 + (0.882 + 0.469i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.615 - 0.788i)T \)
43 \( 1 + (-0.882 - 0.469i)T \)
47 \( 1 + (0.615 + 0.788i)T \)
53 \( 1 + (0.104 + 0.994i)T \)
59 \( 1 + (-0.0348 + 0.999i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (-0.104 + 0.994i)T \)
79 \( 1 + (0.559 - 0.829i)T \)
83 \( 1 + (-0.848 + 0.529i)T \)
89 \( 1 + (-0.913 + 0.406i)T \)
97 \( 1 + (-0.241 - 0.970i)T \)
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

−21.50592012980208515880968805678, −21.1165119622732562909244036149, −20.03550803904715061495680829104, −19.161132047678868710582707724917, −18.47973884695144173294669803262, −18.07460923610454699977128655205, −16.25217199032625130793432799909, −15.83816562725044083769582521866, −14.95957591316548593426973168930, −14.22506265145169737244370700671, −13.593572230999430725744159664449, −12.41207493903791052997526242751, −11.87779527687662969728927432702, −11.13079760975174094218495543858, −10.10613547000271944765945528867, −9.57630846514996335596120542047, −8.13264652699279991723894904292, −7.08942771457817077611113365308, −6.274106950971449308620222060988, −5.48680366888316104147647447740, −4.47441498283120152114349521991, −3.39582226207384950739614115123, −2.576175767043000156634665142471, −1.89933409229880036296263541983, −0.025973203432225034603527986281, 1.246359468223173604007915248637, 2.74871284628426120904373255747, 3.6920390377977393215754325321, 4.559168376903821857689770474186, 5.35635696780216107319070421496, 6.1711124531068402080570072095, 7.386278983590777794869883553109, 7.97501663881136433874483265198, 8.77136279175136280797552330991, 10.20766117774149622388761677857, 10.86781579094870859650108445157, 12.07628668678113183594873073136, 12.74067472667370211906095131773, 13.44659120451883612032749637527, 13.944763626826196350309940288303, 15.346461455334843233433401379608, 15.710541362618398393164584252721, 16.5870263882708503956654210103, 17.26804374476140508746816041137, 17.98662478829321615286996777339, 19.56962447405198724327411548559, 20.08022025435122183879724898081, 20.77968632871489772053245510518, 21.53527887735685542325627064811, 22.41090494710210844463656296468

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