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Properties

Label 1-199-199.180-r0-0-0
Degree $1$
Conductor $199$
Sign $-0.337 - 0.941i$
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  + (−0.939 + 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.5 + 0.866i)5-s + (0.173 + 0.984i)6-s + (−0.939 + 0.342i)7-s + (−0.5 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.173 − 0.984i)10-s + 11-s + (−0.5 − 0.866i)12-s + (0.173 − 0.984i)13-s + (0.766 − 0.642i)14-s + (0.766 + 0.642i)15-s + (0.173 − 0.984i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.5 + 0.866i)5-s + (0.173 + 0.984i)6-s + (−0.939 + 0.342i)7-s + (−0.5 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.173 − 0.984i)10-s + 11-s + (−0.5 − 0.866i)12-s + (0.173 − 0.984i)13-s + (0.766 − 0.642i)14-s + (0.766 + 0.642i)15-s + (0.173 − 0.984i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $-0.337 - 0.941i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (180, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ -0.337 - 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2602064429 - 0.3698639595i\)
\(L(\frac12)\) \(\approx\) \(0.2602064429 - 0.3698639595i\)
\(L(1)\) \(\approx\) \(0.5420877783 - 0.1490366052i\)
\(L(1)\) \(\approx\) \(0.5420877783 - 0.1490366052i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (-0.939 + 0.342i)T \)
3 \( 1 + (0.173 - 0.984i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.939 + 0.342i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (-0.939 - 0.342i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (0.766 + 0.642i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (0.766 - 0.642i)T \)
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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

−27.24488903069780272590434427735, −26.463500695131596166381529469379, −25.635838581665373000953503667219, −24.704466106428613528746434423445, −23.40511633677397451746411408285, −22.16836696973040936156430189171, −21.31983740359372788202995765697, −20.30090425158232080086412572517, −19.6948366337237303190366702830, −18.9787694772921463804279473502, −17.18403605519766267481109155325, −16.66870796047884987135817463919, −16.00231293659410264699501280733, −14.96563581742665393341872658643, −13.42175792985084308805235902048, −12.15172563246481903516932603716, −11.32482237872471209615425411490, −10.13335321642781369416385143099, −9.212656298302215723628807689057, −8.69938076256438758259556327199, −7.30388712887601707846995988767, −5.934895720802965377720934580024, −4.04134162559573407000460659532, −3.58559926268522705520339284363, −1.62405030684304757742813987429, 0.4560658999508312675674534895, 2.31783577312410772465530437247, 3.283023273601253480527484004481, 5.74897221241787340397005960153, 6.80389632760800882160765145433, 7.23253876659243165606741480004, 8.57287984702979291166561689694, 9.4293759695292154044621387846, 10.84541447835448591354537979270, 11.68926823813743543427415625225, 12.82163850422195807516895000305, 14.18458885828862353311926238658, 15.09551315948018893917731091957, 16.027683589476655613942437369549, 17.262787328673629758051591463094, 18.22067587927505861695769878049, 18.81881093209300908464102497803, 19.73195076339728349967611543953, 20.232567435587239418664513530002, 22.350619375375601948140503701837, 22.85237250165906767555018323515, 24.125718310212403592521587668823, 24.92100887429584332296122746699, 25.72245398170140351019845299872, 26.42167212517852500761056198580

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