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Properties

Label 1-187-187.112-r0-0-0
Degree $1$
Conductor $187$
Sign $0.207 - 0.978i$
Analytic cond. $0.868424$
Root an. cond. $0.868424$
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  + (−0.156 − 0.987i)2-s + (0.996 − 0.0784i)3-s + (−0.951 + 0.309i)4-s + (−0.522 + 0.852i)5-s + (−0.233 − 0.972i)6-s + (0.0784 − 0.996i)7-s + (0.453 + 0.891i)8-s + (0.987 − 0.156i)9-s + (0.923 + 0.382i)10-s + (−0.923 + 0.382i)12-s + (0.587 − 0.809i)13-s + (−0.996 + 0.0784i)14-s + (−0.453 + 0.891i)15-s + (0.809 − 0.587i)16-s + (−0.309 − 0.951i)18-s + (0.891 − 0.453i)19-s + ⋯
L(s)  = 1  + (−0.156 − 0.987i)2-s + (0.996 − 0.0784i)3-s + (−0.951 + 0.309i)4-s + (−0.522 + 0.852i)5-s + (−0.233 − 0.972i)6-s + (0.0784 − 0.996i)7-s + (0.453 + 0.891i)8-s + (0.987 − 0.156i)9-s + (0.923 + 0.382i)10-s + (−0.923 + 0.382i)12-s + (0.587 − 0.809i)13-s + (−0.996 + 0.0784i)14-s + (−0.453 + 0.891i)15-s + (0.809 − 0.587i)16-s + (−0.309 − 0.951i)18-s + (0.891 − 0.453i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.207 - 0.978i$
Analytic conductor: \(0.868424\)
Root analytic conductor: \(0.868424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 187,\ (0:\ ),\ 0.207 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.009805978 - 0.8182962861i\)
\(L(\frac12)\) \(\approx\) \(1.009805978 - 0.8182962861i\)
\(L(1)\) \(\approx\) \(1.052162248 - 0.5486358506i\)
\(L(1)\) \(\approx\) \(1.052162248 - 0.5486358506i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.156 - 0.987i)T \)
3 \( 1 + (0.996 - 0.0784i)T \)
5 \( 1 + (-0.522 + 0.852i)T \)
7 \( 1 + (0.0784 - 0.996i)T \)
13 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (0.891 - 0.453i)T \)
23 \( 1 + (0.382 - 0.923i)T \)
29 \( 1 + (0.649 + 0.760i)T \)
31 \( 1 + (-0.233 + 0.972i)T \)
37 \( 1 + (-0.760 + 0.649i)T \)
41 \( 1 + (-0.649 + 0.760i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (0.156 + 0.987i)T \)
59 \( 1 + (0.891 + 0.453i)T \)
61 \( 1 + (-0.972 + 0.233i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.852 - 0.522i)T \)
73 \( 1 + (-0.649 - 0.760i)T \)
79 \( 1 + (-0.852 + 0.522i)T \)
83 \( 1 + (0.987 + 0.156i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.972 + 0.233i)T \)
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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

−27.275664936805784182432045595780, −26.24247305262896048840171260185, −25.39493329062377763727559776612, −24.588732310459563877541751637856, −24.03535678419140587132661909673, −22.83481388639324796603742969780, −21.54272830055345688103092312491, −20.72038557746253860327117231119, −19.365274908481579832209175330, −18.83957028120488332940536458934, −17.66848942888865559936403431064, −16.23351438875838913617773094714, −15.76726428423893774381548264558, −14.84608299087535811409130046650, −13.79652895760558880819477706507, −12.89474916859486451233561752779, −11.69128948341320259586985779359, −9.71377016245142070103589634298, −8.98712909069849010879349716666, −8.25751306063295193729808654551, −7.31345637997811515493521054269, −5.79929933338779458598233956945, −4.63981282062759968642081193891, −3.53642525774102492301854557689, −1.58891502233067146850754367597, 1.21442722953043757555876064849, 2.90538810658963533931481576569, 3.485169678561778048109144021613, 4.672531228857975026642051063015, 6.92791320854478962592124172079, 7.87253375457397570573104528655, 8.834174914510017197470563720812, 10.258632658717156532748177609534, 10.69580450418737818252208850313, 12.054308383394894154970530247018, 13.27121195567152044879943652032, 13.99095966952238496821796866099, 14.903269200696519846485000403173, 16.17955673545526884835444241815, 17.78374309522186522592738292607, 18.46602523740848299475484896663, 19.52702922235561410091824586610, 20.11727245759200412067312386129, 20.88442146681965346454716820818, 22.09234040043875627870482104706, 22.98865801824447540880860007695, 23.90464898794209479479023902700, 25.42249167243968583510374240230, 26.35273806508901500462902729143, 26.90762849966065879826647360908

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