↑

Properties

Label 1-828-828.83-r1-0-0
Degree $1$
Conductor $828$
Sign $-0.619 - 0.785i$
Analytic cond. $88.9809$
Root an. cond. $88.9809$
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  + (0.235 − 0.971i)5-s + (−0.327 + 0.945i)7-s + (−0.0475 + 0.998i)11-s + (−0.327 − 0.945i)13-s + (0.415 + 0.909i)17-s + (0.415 − 0.909i)19-s + (−0.888 − 0.458i)25-s + (−0.580 + 0.814i)29-s + (−0.928 − 0.371i)31-s + (0.841 + 0.540i)35-s + (0.959 + 0.281i)37-s + (−0.235 + 0.971i)41-s + (0.928 − 0.371i)43-s + (−0.5 − 0.866i)47-s + (−0.786 − 0.618i)49-s + ⋯
L(s)  = 1  + (0.235 − 0.971i)5-s + (−0.327 + 0.945i)7-s + (−0.0475 + 0.998i)11-s + (−0.327 − 0.945i)13-s + (0.415 + 0.909i)17-s + (0.415 − 0.909i)19-s + (−0.888 − 0.458i)25-s + (−0.580 + 0.814i)29-s + (−0.928 − 0.371i)31-s + (0.841 + 0.540i)35-s + (0.959 + 0.281i)37-s + (−0.235 + 0.971i)41-s + (0.928 − 0.371i)43-s + (−0.5 − 0.866i)47-s + (−0.786 − 0.618i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.619 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.619 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(828\)    =    \(2^{2} \cdot 3^{2} \cdot 23\)
Sign: $-0.619 - 0.785i$
Analytic conductor: \(88.9809\)
Root analytic conductor: \(88.9809\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{828} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 828,\ (1:\ ),\ -0.619 - 0.785i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4001358963 - 0.8253382393i\)
\(L(\frac12)\) \(\approx\) \(0.4001358963 - 0.8253382393i\)
\(L(1)\) \(\approx\) \(0.9350521189 - 0.1156279487i\)
\(L(1)\) \(\approx\) \(0.9350521189 - 0.1156279487i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 \)
good5 \( 1 + (0.235 - 0.971i)T \)
7 \( 1 + (-0.327 + 0.945i)T \)
11 \( 1 + (-0.0475 + 0.998i)T \)
13 \( 1 + (-0.327 - 0.945i)T \)
17 \( 1 + (0.415 + 0.909i)T \)
19 \( 1 + (0.415 - 0.909i)T \)
29 \( 1 + (-0.580 + 0.814i)T \)
31 \( 1 + (-0.928 - 0.371i)T \)
37 \( 1 + (0.959 + 0.281i)T \)
41 \( 1 + (-0.235 + 0.971i)T \)
43 \( 1 + (0.928 - 0.371i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.654 - 0.755i)T \)
59 \( 1 + (-0.327 - 0.945i)T \)
61 \( 1 + (0.786 - 0.618i)T \)
67 \( 1 + (0.0475 + 0.998i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (0.415 - 0.909i)T \)
79 \( 1 + (0.981 + 0.189i)T \)
83 \( 1 + (-0.235 - 0.971i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (-0.723 - 0.690i)T \)
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

−22.37951271800460541406937844127, −21.45263499220199646609860505252, −20.77754661674467593765187144385, −19.75329983795503784631662505067, −18.93299254953587778637427104692, −18.48128663035582538144823865651, −17.415521162230168596936367245, −16.54316561162634564841369420071, −16.05375896665264682518365890630, −14.7659765905022839402247669036, −13.99576803546680430493775355488, −13.72154100314538923059441168350, −12.47651654484956072367808636594, −11.37424276222817945710346300347, −10.8787345723563579913963937521, −9.861547925299408520003734284625, −9.289589058408011312272193448713, −7.83405186336393570896383941937, −7.242523289365197113073693362806, −6.34730985181650509573906283325, −5.527166541423955715741397267121, −4.130526209853923281011698169577, −3.39813469788025910177514468597, −2.4224092530852333318372772157, −1.09451477275455521069067291293, 0.21206738827590890392206635667, 1.549989256109399403285380506957, 2.48578254780025282355458594246, 3.65799803155595016373745636074, 4.93964720588280422428703042533, 5.41154301460369502025945321851, 6.42966899431695355128238489626, 7.63131776737502745773058751585, 8.418120898524093999189882915112, 9.39692791150460573100591186017, 9.837832482975082344752158321825, 11.069244068967442591633575952604, 12.148135379171990391352055123379, 12.76158630398251596708568185952, 13.18491347916957245405824039019, 14.644654643994402580706932282733, 15.21416544230198605128806567837, 16.03701062472360825985076231323, 16.900736419055913434079913361659, 17.70702385838122023031165158119, 18.33227970963080345814145903048, 19.492966948674254403361553288170, 20.11088175807336569649345097825, 20.79107899321335000467668666208, 21.84754041731744848794490532177

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