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Properties

Label 1-191-191.73-r1-0-0
Degree $1$
Conductor $191$
Sign $0.954 - 0.299i$
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  + (0.746 + 0.665i)2-s + (−0.340 + 0.940i)3-s + (0.115 + 0.993i)4-s + (−0.677 − 0.735i)5-s + (−0.879 + 0.475i)6-s + (−0.309 − 0.951i)7-s + (−0.574 + 0.818i)8-s + (−0.768 − 0.639i)9-s + (−0.0165 − 0.999i)10-s + (0.986 − 0.164i)11-s + (−0.973 − 0.229i)12-s + (−0.277 + 0.960i)13-s + (0.401 − 0.915i)14-s + (0.922 − 0.386i)15-s + (−0.973 + 0.229i)16-s + (0.0495 − 0.998i)17-s + ⋯
L(s)  = 1  + (0.746 + 0.665i)2-s + (−0.340 + 0.940i)3-s + (0.115 + 0.993i)4-s + (−0.677 − 0.735i)5-s + (−0.879 + 0.475i)6-s + (−0.309 − 0.951i)7-s + (−0.574 + 0.818i)8-s + (−0.768 − 0.639i)9-s + (−0.0165 − 0.999i)10-s + (0.986 − 0.164i)11-s + (−0.973 − 0.229i)12-s + (−0.277 + 0.960i)13-s + (0.401 − 0.915i)14-s + (0.922 − 0.386i)15-s + (−0.973 + 0.229i)16-s + (0.0495 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.954 - 0.299i$
Analytic conductor: \(20.5258\)
Root analytic conductor: \(20.5258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (1:\ ),\ 0.954 - 0.299i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.407000401 - 0.2156561511i\)
\(L(\frac12)\) \(\approx\) \(1.407000401 - 0.2156561511i\)
\(L(1)\) \(\approx\) \(1.065820760 + 0.3692421644i\)
\(L(1)\) \(\approx\) \(1.065820760 + 0.3692421644i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (0.746 + 0.665i)T \)
3 \( 1 + (-0.340 + 0.940i)T \)
5 \( 1 + (-0.677 - 0.735i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (0.986 - 0.164i)T \)
13 \( 1 + (-0.277 + 0.960i)T \)
17 \( 1 + (0.0495 - 0.998i)T \)
19 \( 1 + (0.0165 - 0.999i)T \)
23 \( 1 + (0.601 - 0.799i)T \)
29 \( 1 + (0.518 - 0.854i)T \)
31 \( 1 + (-0.245 - 0.969i)T \)
37 \( 1 + (-0.245 + 0.969i)T \)
41 \( 1 + (0.401 + 0.915i)T \)
43 \( 1 + (0.991 + 0.131i)T \)
47 \( 1 + (-0.894 - 0.446i)T \)
53 \( 1 + (0.148 - 0.988i)T \)
59 \( 1 + (0.997 + 0.0660i)T \)
61 \( 1 + (-0.965 - 0.261i)T \)
67 \( 1 + (-0.934 + 0.355i)T \)
71 \( 1 + (0.213 - 0.976i)T \)
73 \( 1 + (-0.701 - 0.712i)T \)
79 \( 1 + (-0.518 - 0.854i)T \)
83 \( 1 + (-0.601 - 0.799i)T \)
89 \( 1 + (-0.431 - 0.901i)T \)
97 \( 1 + (0.371 + 0.928i)T \)
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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

−27.39572267489780488180071486149, −25.52879702466409092026637148438, −24.85728900764182335706957150418, −23.82729291750374861389719644311, −22.83820098962124630248701020526, −22.44119874640102576631467857714, −21.43707110799299374201894894668, −19.75793905627268809318574668098, −19.42922748457252776965792578201, −18.5216969812360117387643159054, −17.56981260494713167513486868244, −15.92664948954264507163538858762, −14.85058060923266032577310072690, −14.18693774053036251818551251786, −12.59736677519727112439753521708, −12.3490481375036005468071850716, −11.292327531869961350976383408421, −10.35844740975540890941089921313, −8.780905552994211917036772139331, −7.342616481699579656710407028040, −6.27558482769172466762049272162, −5.44176414021567315796591998084, −3.70537728590813361103524543099, −2.69034035381624527259173863733, −1.36253188806866711208730159301, 0.42765440062385300202610298740, 3.19076541708176417948672336701, 4.393694627629111787766903114804, 4.661368405820900760433687827427, 6.30316072572137535054047541551, 7.2669998395339696202068937608, 8.70680502846350771070446230017, 9.56869495940241741650442742582, 11.33421608468652789211822958305, 11.81129191652513545113457327853, 13.18723149271648307020239390640, 14.21900098841022206973773421833, 15.18465402345288666806971046634, 16.30945436468566786129381777138, 16.629061387521403821230103006243, 17.48187391854840707708048564309, 19.45950701510170738908514813888, 20.4104512177968546950155397193, 21.14653597080033675113823258471, 22.33693207941452758910161767626, 22.92479694345509076923419664312, 23.867387601621827830125540136916, 24.64230677565400261769310454284, 25.98110963577019062595791017960, 26.795676136220038213930446884129

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