↑

Properties

Label 1-837-837.304-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.179 - 0.983i$
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  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (0.766 + 0.642i)5-s + (−0.939 − 0.342i)7-s + (−0.5 − 0.866i)8-s + 10-s + (0.173 + 0.984i)11-s + (0.173 − 0.984i)13-s + (−0.939 + 0.342i)14-s + (−0.939 − 0.342i)16-s + (−0.5 − 0.866i)17-s + 19-s + (0.766 − 0.642i)20-s + (0.766 + 0.642i)22-s + (0.173 − 0.984i)23-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (0.766 + 0.642i)5-s + (−0.939 − 0.342i)7-s + (−0.5 − 0.866i)8-s + 10-s + (0.173 + 0.984i)11-s + (0.173 − 0.984i)13-s + (−0.939 + 0.342i)14-s + (−0.939 − 0.342i)16-s + (−0.5 − 0.866i)17-s + 19-s + (0.766 − 0.642i)20-s + (0.766 + 0.642i)22-s + (0.173 − 0.984i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.179 - 0.983i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (304, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.179 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.425132650 - 1.708997934i\)
\(L(\frac12)\) \(\approx\) \(1.425132650 - 1.708997934i\)
\(L(1)\) \(\approx\) \(1.429448530 - 0.7631405127i\)
\(L(1)\) \(\approx\) \(1.429448530 - 0.7631405127i\)

Euler product

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

−22.086240443824097703392189802609, −21.80085248707657396008397606967, −21.13706102217646039643682697751, −20.09199409278742709584706431719, −19.258600562847734422622012477728, −18.18713356974534881509282361656, −17.30045737125431329947953944734, −16.48330210341390930930864524464, −16.09832245197061967578930743050, −15.18657834406595870056947356458, −13.96134432474942976677490209516, −13.640689683770265257692855256168, −12.81809228743509105714803497919, −12.03797629654760027806445295690, −11.15179582636422573471490091857, −9.785460032220758469820616442497, −8.977329186581200243268183580501, −8.36991725792592071410311770844, −7.020340403012963544082072367475, −6.215417213445044680353222581434, −5.688709374455508246813040533222, −4.68291070198667910635355136189, −3.61865413735261401032124210970, −2.76719931664865017395969309201, −1.449068026748136381028513312366, 0.79770529216834852446580831969, 2.24630886439054561347268935151, 2.87655460048826668785445571261, 3.81782673407027284505076562539, 4.94352644442371041390160785481, 5.82731654044403234509519762544, 6.69949546838204104751844792582, 7.33415763176721464588382388584, 9.12040452810470390767319303546, 9.85964742260114537520398906154, 10.3769859997504392830696727632, 11.23561434674462329497060771685, 12.39847626061000742132914800038, 12.90794362876459455307223840110, 13.85960850377741194734539993878, 14.30183305681703183123389363689, 15.45882353361456662975486802311, 15.923709456754805962616118900266, 17.33863140897328043465048356236, 18.038945852601632144635031719485, 18.83300845228793872865393096357, 19.748701366592574783612192614081, 20.44265240264791269888037478952, 21.02432108241821251518266557508, 22.257042749949696324678391222063

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