↑

Properties

Label 1-837-837.182-r0-0-0
Degree $1$
Conductor $837$
Sign $0.664 - 0.746i$
Analytic cond. $3.88701$
Root an. cond. $3.88701$
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  + (0.241 − 0.970i)2-s + (−0.882 − 0.469i)4-s + (0.939 − 0.342i)5-s + (0.990 + 0.139i)7-s + (−0.669 + 0.743i)8-s + (−0.104 − 0.994i)10-s + (−0.615 + 0.788i)11-s + (0.241 + 0.970i)13-s + (0.374 − 0.927i)14-s + (0.559 + 0.829i)16-s + (−0.978 − 0.207i)17-s + (0.913 + 0.406i)19-s + (−0.990 − 0.139i)20-s + (0.615 + 0.788i)22-s + (0.990 − 0.139i)23-s + ⋯
L(s)  = 1  + (0.241 − 0.970i)2-s + (−0.882 − 0.469i)4-s + (0.939 − 0.342i)5-s + (0.990 + 0.139i)7-s + (−0.669 + 0.743i)8-s + (−0.104 − 0.994i)10-s + (−0.615 + 0.788i)11-s + (0.241 + 0.970i)13-s + (0.374 − 0.927i)14-s + (0.559 + 0.829i)16-s + (−0.978 − 0.207i)17-s + (0.913 + 0.406i)19-s + (−0.990 − 0.139i)20-s + (0.615 + 0.788i)22-s + (0.990 − 0.139i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.664 - 0.746i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.664 - 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.768491250 - 0.7933486606i\)
\(L(\frac12)\) \(\approx\) \(1.768491250 - 0.7933486606i\)
\(L(1)\) \(\approx\) \(1.274806744 - 0.5526455088i\)
\(L(1)\) \(\approx\) \(1.274806744 - 0.5526455088i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.241 - 0.970i)T \)
5 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (0.990 + 0.139i)T \)
11 \( 1 + (-0.615 + 0.788i)T \)
13 \( 1 + (0.241 + 0.970i)T \)
17 \( 1 + (-0.978 - 0.207i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (0.990 - 0.139i)T \)
29 \( 1 + (-0.241 + 0.970i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.997 + 0.0697i)T \)
43 \( 1 + (-0.961 + 0.275i)T \)
47 \( 1 + (-0.438 + 0.898i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.961 - 0.275i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.978 + 0.207i)T \)
73 \( 1 + (0.978 - 0.207i)T \)
79 \( 1 + (-0.848 - 0.529i)T \)
83 \( 1 + (-0.241 + 0.970i)T \)
89 \( 1 + (-0.978 + 0.207i)T \)
97 \( 1 + (-0.615 + 0.788i)T \)
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

−22.38086172955643555121413755172, −21.44198894049987149300267274771, −21.06894519477338957868069823035, −19.94999474871031539997815492483, −18.505749096604062488450123198301, −18.16562283078945844569707752250, −17.37915720228161910990042448703, −16.79700242190884631812017323542, −15.58315440165627878597440196776, −15.09986786085440937246430400198, −14.158520181592117258307137551237, −13.39637177621982834367511006631, −13.07586829664534631212720204916, −11.54701052797660758967021385212, −10.75415485659260154672615870853, −9.78117022374750904176939335619, −8.77288920465094328430411514414, −8.068182390459196561170276810, −7.18754772316410966335768837744, −6.20502257086089208015154308386, −5.401341305733630395292662311692, −4.82584748681320686571208176247, −3.458946245825252870177690866521, −2.48210914114001687287814630618, −0.94221841168113526850724012776, 1.27747960665914414494263937036, 1.95705138152873459313433308801, 2.81691825547533779755776658070, 4.30963784366756421969052492006, 4.9193095413165262692221824919, 5.65543755323865775089488672088, 6.907289587466441847905331990852, 8.166183500789788949437973577152, 9.16772943737857674033581044000, 9.582819870047597816033730840875, 10.78806003120078246924322900384, 11.24763147416043417426213899595, 12.35909519253039803188769180140, 12.97949452114591513209192794017, 13.92203132297407140323622990928, 14.39381834902057245720037672210, 15.38913150006495477471798756561, 16.6126917957203575681947584889, 17.59236270082351422931568864048, 18.1186652501176053741940727140, 18.687175071413906871608455767949, 20.02755972764635645273345151090, 20.497832898773133227765121209062, 21.28846698767189776297300137831, 21.66831170497511472651831997355

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