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Properties

Label 1-199-199.72-r0-0-0
Degree $1$
Conductor $199$
Sign $0.993 + 0.111i$
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  + (−0.386 + 0.922i)2-s + (−0.745 − 0.666i)3-s + (−0.701 − 0.712i)4-s + (0.981 + 0.189i)5-s + (0.902 − 0.429i)6-s + (−0.999 + 0.0317i)7-s + (0.928 − 0.371i)8-s + (0.110 + 0.993i)9-s + (−0.553 + 0.832i)10-s + (−0.959 + 0.281i)11-s + (0.0475 + 0.998i)12-s + (0.991 − 0.126i)13-s + (0.356 − 0.934i)14-s + (−0.605 − 0.795i)15-s + (−0.0158 + 0.999i)16-s + (0.235 − 0.971i)17-s + ⋯
L(s)  = 1  + (−0.386 + 0.922i)2-s + (−0.745 − 0.666i)3-s + (−0.701 − 0.712i)4-s + (0.981 + 0.189i)5-s + (0.902 − 0.429i)6-s + (−0.999 + 0.0317i)7-s + (0.928 − 0.371i)8-s + (0.110 + 0.993i)9-s + (−0.553 + 0.832i)10-s + (−0.959 + 0.281i)11-s + (0.0475 + 0.998i)12-s + (0.991 − 0.126i)13-s + (0.356 − 0.934i)14-s + (−0.605 − 0.795i)15-s + (−0.0158 + 0.999i)16-s + (0.235 − 0.971i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.993 + 0.111i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (72, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.993 + 0.111i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7293393035 + 0.04081137994i\)
\(L(\frac12)\) \(\approx\) \(0.7293393035 + 0.04081137994i\)
\(L(1)\) \(\approx\) \(0.7177052335 + 0.1087112961i\)
\(L(1)\) \(\approx\) \(0.7177052335 + 0.1087112961i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (-0.386 + 0.922i)T \)
3 \( 1 + (-0.745 - 0.666i)T \)
5 \( 1 + (0.981 + 0.189i)T \)
7 \( 1 + (-0.999 + 0.0317i)T \)
11 \( 1 + (-0.959 + 0.281i)T \)
13 \( 1 + (0.991 - 0.126i)T \)
17 \( 1 + (0.235 - 0.971i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.266 - 0.963i)T \)
29 \( 1 + (0.678 + 0.734i)T \)
31 \( 1 + (0.950 + 0.312i)T \)
37 \( 1 + (0.173 - 0.984i)T \)
41 \( 1 + (0.805 - 0.592i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (0.472 - 0.881i)T \)
53 \( 1 + (-0.605 + 0.795i)T \)
59 \( 1 + (0.580 + 0.814i)T \)
61 \( 1 + (-0.654 - 0.755i)T \)
67 \( 1 + (-0.327 - 0.945i)T \)
71 \( 1 + (0.356 + 0.934i)T \)
73 \( 1 + (0.967 + 0.251i)T \)
79 \( 1 + (-0.857 - 0.513i)T \)
83 \( 1 + (0.0475 - 0.998i)T \)
89 \( 1 + (0.296 - 0.954i)T \)
97 \( 1 + (-0.204 - 0.978i)T \)
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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

−26.98499845777722110772581927853, −26.09598819114942118015749554291, −25.53511794175107183625019615003, −23.72546458070915873421384885697, −22.79690442835991911431133377560, −21.97107630322873006528846862649, −21.13634211511021275703977469001, −20.61479256955483792688398532769, −19.179895094195897874043744165261, −18.24067948539113085324406350699, −17.399851461990123577792296422284, −16.492338152036809695230012092857, −15.670205740574496429170509224780, −13.81434201688383117113587246302, −13.06010698519383749604732768260, −12.06827919054607805347166777311, −10.843449812411872215892060956675, −10.053510971059203857703307442866, −9.48192132766552873354253713352, −8.195043409766129062322356641226, −6.289047674450180193394668822018, −5.43672884437644009717844174899, −3.982349384837421978360637161320, −2.897884958901159083841283842597, −1.16756741331337820938219691314, 0.89151883475798346012434719544, 2.64350718365053009906997984106, 4.914869974215085280682339802621, 5.842002696442269447023404358598, 6.6012143434257410530374411896, 7.50170314359102554404733388481, 8.9180853611046821104409740367, 10.03663660851962575013254542204, 10.80226695537065678117926059264, 12.56406053895528198955345536889, 13.41402221593394804203436038348, 14.06914239742127027754843519978, 15.83165016004533115258206210028, 16.24525834406392814510148003204, 17.458832156525638967875283952250, 18.24812214299604094957894378482, 18.65900609374872827550524328127, 20.03345848790231821584745475152, 21.54705886371656218787035852336, 22.78356052604416908856955722691, 22.995013360004536625332465052032, 24.284328945276880962590138069600, 25.10705901710350910811537961853, 25.82134920422376772385398041986, 26.61430241107255403954625868444

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