Properties

Label 1-199-199.161-r0-0-0
Degree $1$
Conductor $199$
Sign $-0.683 - 0.730i$
Analytic cond. $0.924152$
Root an. cond. $0.924152$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.356 + 0.934i)2-s + (−0.386 + 0.922i)3-s + (−0.745 + 0.666i)4-s + (0.235 + 0.971i)5-s + (−0.999 − 0.0317i)6-s + (−0.975 + 0.220i)7-s + (−0.888 − 0.458i)8-s + (−0.701 − 0.712i)9-s + (−0.823 + 0.567i)10-s + (0.415 + 0.909i)11-s + (−0.327 − 0.945i)12-s + (0.630 − 0.776i)13-s + (−0.553 − 0.832i)14-s + (−0.987 − 0.158i)15-s + (0.110 − 0.993i)16-s + (−0.995 − 0.0950i)17-s + ⋯
L(s)  = 1  + (0.356 + 0.934i)2-s + (−0.386 + 0.922i)3-s + (−0.745 + 0.666i)4-s + (0.235 + 0.971i)5-s + (−0.999 − 0.0317i)6-s + (−0.975 + 0.220i)7-s + (−0.888 − 0.458i)8-s + (−0.701 − 0.712i)9-s + (−0.823 + 0.567i)10-s + (0.415 + 0.909i)11-s + (−0.327 − 0.945i)12-s + (0.630 − 0.776i)13-s + (−0.553 − 0.832i)14-s + (−0.987 − 0.158i)15-s + (0.110 − 0.993i)16-s + (−0.995 − 0.0950i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $-0.683 - 0.730i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ -0.683 - 0.730i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3242444344 + 0.7476953342i\)
\(L(\frac12)\) \(\approx\) \(-0.3242444344 + 0.7476953342i\)
\(L(1)\) \(\approx\) \(0.3854808812 + 0.7886712721i\)
\(L(1)\) \(\approx\) \(0.3854808812 + 0.7886712721i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (0.356 + 0.934i)T \)
3 \( 1 + (-0.386 + 0.922i)T \)
5 \( 1 + (0.235 + 0.971i)T \)
7 \( 1 + (-0.975 + 0.220i)T \)
11 \( 1 + (0.415 + 0.909i)T \)
13 \( 1 + (0.630 - 0.776i)T \)
17 \( 1 + (-0.995 - 0.0950i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (0.950 - 0.312i)T \)
29 \( 1 + (0.873 - 0.486i)T \)
31 \( 1 + (-0.605 + 0.795i)T \)
37 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (-0.266 + 0.963i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (0.296 - 0.954i)T \)
53 \( 1 + (-0.987 + 0.158i)T \)
59 \( 1 + (0.928 + 0.371i)T \)
61 \( 1 + (-0.959 + 0.281i)T \)
67 \( 1 + (0.723 - 0.690i)T \)
71 \( 1 + (-0.553 + 0.832i)T \)
73 \( 1 + (-0.204 + 0.978i)T \)
79 \( 1 + (0.805 + 0.592i)T \)
83 \( 1 + (-0.327 + 0.945i)T \)
89 \( 1 + (-0.857 - 0.513i)T \)
97 \( 1 + (0.991 + 0.126i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.38664703678525817312490739423, −25.146255026413122474375988514480, −24.05617879587954719878920156067, −23.6291721584740650032850274389, −22.42516186745259301264377034030, −21.69939467951025300105683349638, −20.48213177993066652030215062000, −19.51739939145095631079586504427, −19.0571032625525850512749497118, −17.801522407022708807364825729775, −16.88328941180384829495858502570, −15.82464942514491971063616554572, −13.926727549861450628572435253, −13.38122464198147827515304946002, −12.70395757901337690291874996495, −11.6379517920720759775658853602, −10.83657351864171361118209291269, −9.227630787412698070889664320184, −8.6855093277087744219383512427, −6.7541167752348954794447497296, −5.86224059268311940434137841334, −4.65306858420480044283094620949, −3.250149235190862336078056117530, −1.79112323305518250504249862775, −0.59211667654794423048426839746, 2.99071140149756090901736890557, 3.85276578688113039950275562208, 5.19394617044801645541729699842, 6.30007089530205803328661959570, 6.907912141032357541587051658321, 8.58253763694228618283434117307, 9.672962336425704979480150923633, 10.47144964721436061888429578659, 11.899850885844143332234130197968, 13.03022018567874169443849456341, 14.21857944778151291864085608212, 15.22132493465195926640929301246, 15.65283956917931612807234425972, 16.79100469799009992413678781741, 17.70614218960426470698340291631, 18.53287524599219070686815129355, 20.04127549503838156600372546003, 21.288739049639802896504486700150, 22.22146243862671435990675854682, 22.802313789596011223345251796778, 23.234486152119016703852226938884, 25.145419191320014435559708557275, 25.47821363830114654443967512972, 26.56903975935051234507679684854, 27.11331285268785834735337499052

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