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Properties

Label 1-191-191.31-r1-0-0
Degree $1$
Conductor $191$
Sign $0.748 + 0.663i$
Analytic cond. $20.5258$
Root an. cond. $20.5258$
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  + (−0.986 − 0.164i)2-s + (0.546 − 0.837i)3-s + (0.945 + 0.324i)4-s + (0.945 + 0.324i)5-s + (−0.677 + 0.735i)6-s − 7-s + (−0.879 − 0.475i)8-s + (−0.401 − 0.915i)9-s + (−0.879 − 0.475i)10-s + (−0.245 + 0.969i)11-s + (0.789 − 0.614i)12-s + (0.546 + 0.837i)13-s + (0.986 + 0.164i)14-s + (0.789 − 0.614i)15-s + (0.789 + 0.614i)16-s + (−0.0825 + 0.996i)17-s + ⋯
L(s)  = 1  + (−0.986 − 0.164i)2-s + (0.546 − 0.837i)3-s + (0.945 + 0.324i)4-s + (0.945 + 0.324i)5-s + (−0.677 + 0.735i)6-s − 7-s + (−0.879 − 0.475i)8-s + (−0.401 − 0.915i)9-s + (−0.879 − 0.475i)10-s + (−0.245 + 0.969i)11-s + (0.789 − 0.614i)12-s + (0.546 + 0.837i)13-s + (0.986 + 0.164i)14-s + (0.789 − 0.614i)15-s + (0.789 + 0.614i)16-s + (−0.0825 + 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.748 + 0.663i$
Analytic conductor: \(20.5258\)
Root analytic conductor: \(20.5258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (1:\ ),\ 0.748 + 0.663i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.169284210 + 0.4436039051i\)
\(L(\frac12)\) \(\approx\) \(1.169284210 + 0.4436039051i\)
\(L(1)\) \(\approx\) \(0.8753490001 - 0.03846402956i\)
\(L(1)\) \(\approx\) \(0.8753490001 - 0.03846402956i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (-0.986 - 0.164i)T \)
3 \( 1 + (0.546 - 0.837i)T \)
5 \( 1 + (0.945 + 0.324i)T \)
7 \( 1 - T \)
11 \( 1 + (-0.245 + 0.969i)T \)
13 \( 1 + (0.546 + 0.837i)T \)
17 \( 1 + (-0.0825 + 0.996i)T \)
19 \( 1 + (0.879 - 0.475i)T \)
23 \( 1 + (-0.879 + 0.475i)T \)
29 \( 1 + (-0.789 + 0.614i)T \)
31 \( 1 + (0.401 + 0.915i)T \)
37 \( 1 + (0.401 - 0.915i)T \)
41 \( 1 + (0.986 - 0.164i)T \)
43 \( 1 + (-0.677 + 0.735i)T \)
47 \( 1 + (-0.245 + 0.969i)T \)
53 \( 1 + (-0.245 + 0.969i)T \)
59 \( 1 + (-0.401 - 0.915i)T \)
61 \( 1 + (0.0825 + 0.996i)T \)
67 \( 1 + (-0.0825 - 0.996i)T \)
71 \( 1 + (0.986 - 0.164i)T \)
73 \( 1 + (-0.245 - 0.969i)T \)
79 \( 1 + (0.789 + 0.614i)T \)
83 \( 1 + (0.879 + 0.475i)T \)
89 \( 1 + (0.677 + 0.735i)T \)
97 \( 1 + (-0.401 + 0.915i)T \)
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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

−26.52691158047780852614782375057, −26.002539828672477263586948896610, −25.08427221307696955244984612321, −24.48823854351063376714892779987, −22.72315701474431918796302283372, −21.79067361523839320557623794292, −20.599132528471177182762630313547, −20.26874818597324122352017102984, −18.950448247889060163899231558133, −18.14470042392018440945375594054, −16.76088493110125391071565275866, −16.24346613641414912353355051711, −15.42107052225388397941472114619, −14.0411189385235201621522983755, −13.19702699210462414762914567825, −11.49996931649052964863113264987, −10.228506632554542292669110315156, −9.75441376800907867731167179271, −8.821973423083938681199063888849, −7.85592802693822021700432470286, −6.19782132248650216565654107103, −5.423586036587168372317832234364, −3.38730015248588591466026888937, −2.41446169211980894203149215714, −0.56071117044403861722198546962, 1.38909207617509608048684453998, 2.31625743104674431939409808693, 3.46787164389244061686139642142, 6.03170051327709044293563279313, 6.745034958902002577237561545786, 7.69978185151317255190528676057, 9.15957541026426904868995948288, 9.59467960954420486949571004109, 10.83210220563554931475259336455, 12.25862449048422894731056992411, 13.04070609882841810830372745177, 14.1377861362047168067715988158, 15.36083237371227052007339181948, 16.51746861220866179302005259970, 17.7375774535865509323278996221, 18.17900480881604015669013248921, 19.2422982423650271351371995057, 19.9397530208969778279516524423, 20.9561239435818165949568149914, 21.98913427511604613017724775637, 23.363209402155855895365201779555, 24.47951048466027830240533360221, 25.42990367216526271563060200797, 26.05087631333705367146164513499, 26.38553603326946069009729050036

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