Properties

Label 1-199-199.33-r0-0-0
Degree $1$
Conductor $199$
Sign $0.982 - 0.184i$
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  + (−0.204 − 0.978i)2-s + (0.967 − 0.251i)3-s + (−0.916 + 0.400i)4-s + (−0.888 + 0.458i)5-s + (−0.444 − 0.895i)6-s + (−0.0792 + 0.996i)7-s + (0.580 + 0.814i)8-s + (0.873 − 0.486i)9-s + (0.630 + 0.776i)10-s + (−0.654 + 0.755i)11-s + (−0.786 + 0.618i)12-s + (0.950 + 0.312i)13-s + (0.991 − 0.126i)14-s + (−0.745 + 0.666i)15-s + (0.678 − 0.734i)16-s + (0.981 + 0.189i)17-s + ⋯
L(s)  = 1  + (−0.204 − 0.978i)2-s + (0.967 − 0.251i)3-s + (−0.916 + 0.400i)4-s + (−0.888 + 0.458i)5-s + (−0.444 − 0.895i)6-s + (−0.0792 + 0.996i)7-s + (0.580 + 0.814i)8-s + (0.873 − 0.486i)9-s + (0.630 + 0.776i)10-s + (−0.654 + 0.755i)11-s + (−0.786 + 0.618i)12-s + (0.950 + 0.312i)13-s + (0.991 − 0.126i)14-s + (−0.745 + 0.666i)15-s + (0.678 − 0.734i)16-s + (0.981 + 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.982 - 0.184i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.982 - 0.184i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.130129133 - 0.1050025259i\)
\(L(\frac12)\) \(\approx\) \(1.130129133 - 0.1050025259i\)
\(L(1)\) \(\approx\) \(1.034928445 - 0.2331460647i\)
\(L(1)\) \(\approx\) \(1.034928445 - 0.2331460647i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (-0.204 - 0.978i)T \)
3 \( 1 + (0.967 - 0.251i)T \)
5 \( 1 + (-0.888 + 0.458i)T \)
7 \( 1 + (-0.0792 + 0.996i)T \)
11 \( 1 + (-0.654 + 0.755i)T \)
13 \( 1 + (0.950 + 0.312i)T \)
17 \( 1 + (0.981 + 0.189i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.110 + 0.993i)T \)
29 \( 1 + (0.472 + 0.881i)T \)
31 \( 1 + (-0.701 + 0.712i)T \)
37 \( 1 + (-0.939 - 0.342i)T \)
41 \( 1 + (-0.0158 + 0.999i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (0.902 - 0.429i)T \)
53 \( 1 + (-0.745 - 0.666i)T \)
59 \( 1 + (0.723 + 0.690i)T \)
61 \( 1 + (0.841 - 0.540i)T \)
67 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (0.991 + 0.126i)T \)
73 \( 1 + (0.805 - 0.592i)T \)
79 \( 1 + (-0.975 - 0.220i)T \)
83 \( 1 + (-0.786 - 0.618i)T \)
89 \( 1 + (-0.999 - 0.0317i)T \)
97 \( 1 + (-0.266 - 0.963i)T \)
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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

−26.90135229527823855044639681301, −26.04855841331189027617012693182, −25.15865138853806462643418799623, −24.14989122916896282266331708929, −23.46296494525464273901262383365, −22.57682194648722294344729776343, −20.92038758790800998276837191760, −20.35667431523519030453546936778, −19.03734835687936690109688574618, −18.63381876049552671909864702994, −16.91059973279251419629948585704, −16.17533835190410687377330752216, −15.538575966564111020041834223003, −14.35002364502430382351784207651, −13.621068602600324904880368083575, −12.66176377124914652452959726156, −10.7746534050364033712479956962, −9.871407327416082406604380157897, −8.44605826171973262643259568201, −8.07613506310103841094970295220, −7.093419167661262914584148038934, −5.4969018515455954703533613307, −4.15231632076971551432558990293, −3.45622132325062073780097916084, −0.9660403212515779055446842140, 1.64175241990932467134824997341, 2.90505571336606257749310238337, 3.604257743967648905668411180316, 5.0417963295255738843245339393, 7.049771142670902715204378184722, 8.12118861414974423492136086129, 8.89990798629512205706595787555, 9.964493579629623446589112590910, 11.18468683290919664099686621196, 12.20761212869680565789674544423, 12.96225943513694519667771140771, 14.15020868352477698345729738067, 15.15549389745077653385473846217, 16.00624109131901252673189756808, 17.98281822435478605315992561322, 18.48264300040353193734949432946, 19.34096653299430114670409730954, 20.04714133004594880550275317615, 21.101488121176642550399670946017, 21.80945083287996955104746562881, 23.1497002345867553146599605182, 23.79867931742930115803966808669, 25.51835742491499830905746642341, 25.88950338152931057844402168960, 26.98999319484175551554300671362

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