↑

Properties

Label 1-1684-1684.1343-r1-0-0
Degree $1$
Conductor $1684$
Sign $-0.988 + 0.152i$
Analytic cond. $180.970$
Root an. cond. $180.970$
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  + (0.0448 − 0.998i)3-s + (−0.550 − 0.834i)5-s + (0.0448 + 0.998i)7-s + (−0.995 − 0.0896i)9-s + (0.963 − 0.266i)11-s + (−0.309 + 0.951i)13-s + (−0.858 + 0.512i)15-s + (0.983 + 0.178i)17-s + (−0.691 − 0.722i)19-s + 21-s + (0.936 + 0.351i)23-s + (−0.393 + 0.919i)25-s + (−0.134 + 0.990i)27-s − 29-s + (−0.753 + 0.657i)31-s + ⋯
L(s)  = 1  + (0.0448 − 0.998i)3-s + (−0.550 − 0.834i)5-s + (0.0448 + 0.998i)7-s + (−0.995 − 0.0896i)9-s + (0.963 − 0.266i)11-s + (−0.309 + 0.951i)13-s + (−0.858 + 0.512i)15-s + (0.983 + 0.178i)17-s + (−0.691 − 0.722i)19-s + 21-s + (0.936 + 0.351i)23-s + (−0.393 + 0.919i)25-s + (−0.134 + 0.990i)27-s − 29-s + (−0.753 + 0.657i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.988 + 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.988 + 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1684\)    =    \(2^{2} \cdot 421\)
Sign: $-0.988 + 0.152i$
Analytic conductor: \(180.970\)
Root analytic conductor: \(180.970\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1684} (1343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1684,\ (1:\ ),\ -0.988 + 0.152i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04931042440 - 0.6422551902i\)
\(L(\frac12)\) \(\approx\) \(-0.04931042440 - 0.6422551902i\)
\(L(1)\) \(\approx\) \(0.8378139300 - 0.3142470591i\)
\(L(1)\) \(\approx\) \(0.8378139300 - 0.3142470591i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
421 \( 1 \)
good3 \( 1 + (0.0448 - 0.998i)T \)
5 \( 1 + (-0.550 - 0.834i)T \)
7 \( 1 + (0.0448 + 0.998i)T \)
11 \( 1 + (0.963 - 0.266i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (0.983 + 0.178i)T \)
19 \( 1 + (-0.691 - 0.722i)T \)
23 \( 1 + (0.936 + 0.351i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.753 + 0.657i)T \)
37 \( 1 + (0.963 - 0.266i)T \)
41 \( 1 + (0.691 + 0.722i)T \)
43 \( 1 + (-0.393 - 0.919i)T \)
47 \( 1 + (-0.550 - 0.834i)T \)
53 \( 1 + (0.0448 + 0.998i)T \)
59 \( 1 + (0.134 - 0.990i)T \)
61 \( 1 + (0.393 + 0.919i)T \)
67 \( 1 + (-0.309 - 0.951i)T \)
71 \( 1 + (0.134 + 0.990i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.393 - 0.919i)T \)
83 \( 1 + (0.134 - 0.990i)T \)
89 \( 1 + (-0.753 - 0.657i)T \)
97 \( 1 + (-0.691 + 0.722i)T \)
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

−20.62280651966314159380949102618, −19.78919901262071111670100748702, −19.343995568973896490500192074296, −18.338999881399125196978796275382, −17.38230739710329129927355806351, −16.7491508728074227689642902916, −16.25008449595408297800911331564, −15.01547483745488623167000266483, −14.77410569609229278387380519806, −14.23719206356716151697545693225, −13.08711331717137792105943503258, −12.16183429979604646208575302469, −11.15204747142707356959484582002, −10.83373813168141349535186847371, −9.938979548033876709961958677556, −9.45265654653267854969561368883, −8.158033639529433764754874476639, −7.62980163584668618775292833497, −6.695462808067586770826179170038, −5.799652646366376163799527132787, −4.76325862331676748162654245740, −3.86001037231680860502742075024, −3.5108325579554730773882472796, −2.48986178978507042268225183897, −0.98116110436455687966821994436, 0.1351638853748342820695948603, 1.24951654680515698346489936154, 1.91792015058685130732806964404, 3.03195987474113826701129084620, 4.00347846203125454420861856603, 5.09198259563931179381266669313, 5.7909516232521763048328158245, 6.719055190629881204527720793567, 7.445723575455107601173648786753, 8.37790413095354649946770254791, 8.9941208218173978541794626236, 9.44508303683640421965447617042, 11.21515708115076466897314303869, 11.54927584322017446152407123600, 12.37482681368976598289941284282, 12.78020535800703459629403762387, 13.68562029196912801181369337485, 14.67940700995746920302584591249, 15.0555216147764968953685936989, 16.34223430057349780916388740279, 16.80240504866014797181734983347, 17.49929637430822293872244616446, 18.55066307981172037288094265065, 19.07987463951747258903336610656, 19.577620574002265213169320184136

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