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Properties

Label 1-191-191.125-r0-0-0
Degree $1$
Conductor $191$
Sign $0.688 - 0.724i$
Analytic cond. $0.887000$
Root an. cond. $0.887000$
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  + (−0.0825 − 0.996i)2-s + (−0.879 − 0.475i)3-s + (−0.986 + 0.164i)4-s + (−0.986 + 0.164i)5-s + (−0.401 + 0.915i)6-s + 7-s + (0.245 + 0.969i)8-s + (0.546 + 0.837i)9-s + (0.245 + 0.969i)10-s + (0.789 + 0.614i)11-s + (0.945 + 0.324i)12-s + (−0.879 + 0.475i)13-s + (−0.0825 − 0.996i)14-s + (0.945 + 0.324i)15-s + (0.945 − 0.324i)16-s + (−0.677 + 0.735i)17-s + ⋯
L(s)  = 1  + (−0.0825 − 0.996i)2-s + (−0.879 − 0.475i)3-s + (−0.986 + 0.164i)4-s + (−0.986 + 0.164i)5-s + (−0.401 + 0.915i)6-s + 7-s + (0.245 + 0.969i)8-s + (0.546 + 0.837i)9-s + (0.245 + 0.969i)10-s + (0.789 + 0.614i)11-s + (0.945 + 0.324i)12-s + (−0.879 + 0.475i)13-s + (−0.0825 − 0.996i)14-s + (0.945 + 0.324i)15-s + (0.945 − 0.324i)16-s + (−0.677 + 0.735i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.688 - 0.724i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ 0.688 - 0.724i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6112614518 - 0.2624181331i\)
\(L(\frac12)\) \(\approx\) \(0.6112614518 - 0.2624181331i\)
\(L(1)\) \(\approx\) \(0.6247347628 - 0.2883248900i\)
\(L(1)\) \(\approx\) \(0.6247347628 - 0.2883248900i\)

Euler product

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

−27.27022275227274597052880371698, −26.74315414032333232172285276268, −24.98266810377365325725728209357, −24.28661835748487876995123861750, −23.541292029344575617921693386583, −22.60185786327534130151578780581, −21.914526095120505398019610364028, −20.61511280886444882184678974950, −19.305627783063310046989375454935, −18.21006839690727360247198826544, −17.25709477636899851677913033712, −16.61046381529261254034640564739, −15.524411633783039698989642274733, −14.94539770715841571538654551752, −13.749459490133096152090883269732, −12.14044256839397576879064446965, −11.508095140808047974674674326920, −10.19696529079376421300350681062, −8.950954778527336807304692496007, −7.876531390029379391858020915717, −6.90477856781027278664670705839, −5.559131352532656427165180388062, −4.697488763371106097622756514149, −3.77303145264603832314762177722, −0.805888258702357857172974931864, 1.11704389899231941292120243945, 2.45125132338050240490685564253, 4.40742660714128937286129890534, 4.74829630629034202840719694436, 6.704066089009344750888757210090, 7.77757259186607086857371634482, 8.913083872919789724895787147881, 10.46344899225023723331863761298, 11.26671814173876734633720842073, 11.97913950806669545170127410760, 12.65787145374292500321671787724, 14.110211988759123616833677671356, 15.08924290237290855161891424054, 16.68110947195476420375433011272, 17.58860839998057036326483563970, 18.25607954611852966078358540806, 19.51403375171857527302060098666, 19.882844960285283882571447696289, 21.418787834444759962441384097026, 22.15820161706669607959811816439, 23.073738218217630824980888935793, 23.87093187712861209404342707825, 24.710613585016581998347842539776, 26.61064837151369317259071352966, 27.15650364895767735168194627001

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