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Properties

Label 1-191-191.40-r0-0-0
Degree $1$
Conductor $191$
Sign $0.0186 + 0.999i$
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  + (0.601 + 0.799i)2-s + (0.746 + 0.665i)3-s + (−0.277 + 0.960i)4-s + (0.789 − 0.614i)5-s + (−0.0825 + 0.996i)6-s + (0.309 − 0.951i)7-s + (−0.934 + 0.355i)8-s + (0.115 + 0.993i)9-s + (0.965 + 0.261i)10-s + (−0.879 + 0.475i)11-s + (−0.846 + 0.533i)12-s + (−0.213 + 0.976i)13-s + (0.945 − 0.324i)14-s + (0.997 + 0.0660i)15-s + (−0.846 − 0.533i)16-s + (0.701 − 0.712i)17-s + ⋯
L(s)  = 1  + (0.601 + 0.799i)2-s + (0.746 + 0.665i)3-s + (−0.277 + 0.960i)4-s + (0.789 − 0.614i)5-s + (−0.0825 + 0.996i)6-s + (0.309 − 0.951i)7-s + (−0.934 + 0.355i)8-s + (0.115 + 0.993i)9-s + (0.965 + 0.261i)10-s + (−0.879 + 0.475i)11-s + (−0.846 + 0.533i)12-s + (−0.213 + 0.976i)13-s + (0.945 − 0.324i)14-s + (0.997 + 0.0660i)15-s + (−0.846 − 0.533i)16-s + (0.701 − 0.712i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0186 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0186 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.0186 + 0.999i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ 0.0186 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.443413614 + 1.470638102i\)
\(L(\frac12)\) \(\approx\) \(1.443413614 + 1.470638102i\)
\(L(1)\) \(\approx\) \(1.495296276 + 0.9754905125i\)
\(L(1)\) \(\approx\) \(1.495296276 + 0.9754905125i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (0.601 + 0.799i)T \)
3 \( 1 + (0.746 + 0.665i)T \)
5 \( 1 + (0.789 - 0.614i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-0.879 + 0.475i)T \)
13 \( 1 + (-0.213 + 0.976i)T \)
17 \( 1 + (0.701 - 0.712i)T \)
19 \( 1 + (0.965 - 0.261i)T \)
23 \( 1 + (-0.627 + 0.778i)T \)
29 \( 1 + (-0.768 - 0.639i)T \)
31 \( 1 + (-0.677 - 0.735i)T \)
37 \( 1 + (-0.677 + 0.735i)T \)
41 \( 1 + (0.945 + 0.324i)T \)
43 \( 1 + (-0.518 - 0.854i)T \)
47 \( 1 + (0.431 - 0.901i)T \)
53 \( 1 + (-0.724 - 0.689i)T \)
59 \( 1 + (0.490 - 0.871i)T \)
61 \( 1 + (-0.461 + 0.887i)T \)
67 \( 1 + (0.894 - 0.446i)T \)
71 \( 1 + (-0.956 + 0.293i)T \)
73 \( 1 + (0.991 - 0.131i)T \)
79 \( 1 + (-0.768 + 0.639i)T \)
83 \( 1 + (-0.627 - 0.778i)T \)
89 \( 1 + (0.652 + 0.757i)T \)
97 \( 1 + (0.980 - 0.197i)T \)
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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

−26.78963321863277194398160281115, −25.76361148525944100668812868457, −24.78542978105543555554571057061, −24.11494634353077339373393226244, −22.88744109916708091585213677147, −21.89328928390368873961824284981, −21.1181258489106359438437861885, −20.30178537911455595739895984447, −19.098897244822694284626160447437, −18.35729550471832387449507450819, −17.86404793855265151641753560847, −15.69419515283459962473001194012, −14.61769912745927493040553667170, −14.15044902261147211494602857982, −12.91896661759837315634714707310, −12.394860541330468366323657820375, −10.95959375416669536531020325633, −9.980414647016821412411280785924, −8.888296970140161281265865109, −7.65506997900639411715726125086, −6.02791609018801991557293613748, −5.40147164460545747312608255716, −3.33115166986875616220967648434, −2.63770694458763311809503913938, −1.57114803722613614341843712711, 2.13169345144179608479620032494, 3.6351966158997327364723325691, 4.72830980250936946286130112164, 5.450453695903525084995739599471, 7.20790417284409409705235869712, 7.962347668918180512443735757158, 9.2965362516102441467440837884, 9.98340824778850592020131669807, 11.630801934730387314007735078761, 13.14733645309536378423708024820, 13.77046603506993516221583978997, 14.44251306569708747323363389698, 15.69723154418691722712520888995, 16.49438081544852081214678004583, 17.26427685381803822093248040124, 18.45921951658537402674992246346, 20.21656197275486766166983572930, 20.77223756378743961077647610100, 21.51118141488084124950729732358, 22.53586001837384752962222236447, 23.77070350305568237338830265572, 24.45092155200634811074448836375, 25.57264539363948351623093001372, 26.12922568374663040257620249169, 26.92604134443079925933377340211

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