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Properties

Label 1-191-191.13-r0-0-0
Degree $1$
Conductor $191$
Sign $0.684 - 0.729i$
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  + (0.922 − 0.386i)2-s + (−0.724 + 0.689i)3-s + (0.701 − 0.712i)4-s + (−0.986 + 0.164i)5-s + (−0.401 + 0.915i)6-s + (0.309 − 0.951i)7-s + (0.371 − 0.928i)8-s + (0.0495 − 0.998i)9-s + (−0.846 + 0.533i)10-s + (0.789 + 0.614i)11-s + (−0.0165 + 0.999i)12-s + (0.991 + 0.131i)13-s + (−0.0825 − 0.996i)14-s + (0.601 − 0.799i)15-s + (−0.0165 − 0.999i)16-s + (0.115 − 0.993i)17-s + ⋯
L(s)  = 1  + (0.922 − 0.386i)2-s + (−0.724 + 0.689i)3-s + (0.701 − 0.712i)4-s + (−0.986 + 0.164i)5-s + (−0.401 + 0.915i)6-s + (0.309 − 0.951i)7-s + (0.371 − 0.928i)8-s + (0.0495 − 0.998i)9-s + (−0.846 + 0.533i)10-s + (0.789 + 0.614i)11-s + (−0.0165 + 0.999i)12-s + (0.991 + 0.131i)13-s + (−0.0825 − 0.996i)14-s + (0.601 − 0.799i)15-s + (−0.0165 − 0.999i)16-s + (0.115 − 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.684 - 0.729i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ 0.684 - 0.729i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.338416493 - 0.5792717121i\)
\(L(\frac12)\) \(\approx\) \(1.338416493 - 0.5792717121i\)
\(L(1)\) \(\approx\) \(1.304485989 - 0.2938015649i\)
\(L(1)\) \(\approx\) \(1.304485989 - 0.2938015649i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (0.922 - 0.386i)T \)
3 \( 1 + (-0.724 + 0.689i)T \)
5 \( 1 + (-0.986 + 0.164i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (0.789 + 0.614i)T \)
13 \( 1 + (0.991 + 0.131i)T \)
17 \( 1 + (0.115 - 0.993i)T \)
19 \( 1 + (-0.846 - 0.533i)T \)
23 \( 1 + (0.997 - 0.0660i)T \)
29 \( 1 + (-0.956 + 0.293i)T \)
31 \( 1 + (0.546 + 0.837i)T \)
37 \( 1 + (0.546 - 0.837i)T \)
41 \( 1 + (-0.0825 + 0.996i)T \)
43 \( 1 + (-0.213 - 0.976i)T \)
47 \( 1 + (-0.999 - 0.0330i)T \)
53 \( 1 + (-0.340 + 0.940i)T \)
59 \( 1 + (-0.627 + 0.778i)T \)
61 \( 1 + (-0.909 + 0.416i)T \)
67 \( 1 + (0.980 + 0.197i)T \)
71 \( 1 + (-0.518 - 0.854i)T \)
73 \( 1 + (-0.277 + 0.960i)T \)
79 \( 1 + (-0.956 - 0.293i)T \)
83 \( 1 + (0.997 + 0.0660i)T \)
89 \( 1 + (0.863 + 0.504i)T \)
97 \( 1 + (-0.934 + 0.355i)T \)
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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

−27.48573827556332753435651870339, −25.92812811499934031071330839932, −24.856108635005395172915074355708, −24.27286864696001090949919633954, −23.41132295962302965115438381705, −22.7015722112508950660784970825, −21.77471063184726033563239479031, −20.77984908153075920632827754557, −19.34944141387730685357654220480, −18.69804006918002188309268233717, −17.228266092759590028045162772894, −16.503865097411188998605338064557, −15.44010835728835527576526693447, −14.62773538088749960468563388646, −13.19692772070652105447749609268, −12.49926554634303596949462686362, −11.48994390031188431925117030755, −11.07731735087500861171650146797, −8.547730201162868905421907156143, −7.9317891685751831365704413809, −6.49090827868312009018095919866, −5.849245622621545382991870918526, −4.57832718817152383855673338897, −3.38122098174780747106447283513, −1.66384539602937138884114082964, 1.08423837781774638702613666488, 3.31196676358670829805119137650, 4.21330004010191355729276311585, 4.87043132244986228006354310118, 6.47540590567373216052422313599, 7.25005844771564088461023638641, 9.15337650609388075125852118898, 10.54859748646054634679611176468, 11.17320892984631281650452136805, 11.90000994653593274299574019804, 13.0705535730740782777156974356, 14.399324981860822118194267340, 15.174110574134254144108724918019, 16.11978450406763252279510343918, 16.96836666899025394138325527329, 18.37542781332057802856317770551, 19.69565895451035527934267496540, 20.436080408152582493146721230146, 21.25237362162521508723027646569, 22.431660332439315003287172994598, 23.29174892652475356653515352089, 23.385274921673852946836657954118, 24.75808139413034170124239579353, 26.166730108776777405927968391444, 27.33028975388262831739662996304

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