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Properties

Label 1-189-189.2-r1-0-0
Degree $1$
Conductor $189$
Sign $0.941 - 0.337i$
Analytic cond. $20.3108$
Root an. cond. $20.3108$
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  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.939 − 0.342i)5-s + (0.5 + 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.939 + 0.342i)11-s + (−0.939 + 0.342i)13-s + (0.766 − 0.642i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.766 + 0.642i)20-s + (0.173 − 0.984i)22-s + (−0.173 + 0.984i)23-s + (0.766 − 0.642i)25-s + (0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.939 − 0.342i)5-s + (0.5 + 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.939 + 0.342i)11-s + (−0.939 + 0.342i)13-s + (0.766 − 0.642i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.766 + 0.642i)20-s + (0.173 − 0.984i)22-s + (−0.173 + 0.984i)23-s + (0.766 − 0.642i)25-s + (0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.941 - 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.941 - 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.941 - 0.337i$
Analytic conductor: \(20.3108\)
Root analytic conductor: \(20.3108\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (1:\ ),\ 0.941 - 0.337i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.735831488 - 0.3014247479i\)
\(L(\frac12)\) \(\approx\) \(1.735831488 - 0.3014247479i\)
\(L(1)\) \(\approx\) \(1.061693303 - 0.3425501683i\)
\(L(1)\) \(\approx\) \(1.061693303 - 0.3425501683i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.173 - 0.984i)T \)
5 \( 1 + (0.939 - 0.342i)T \)
11 \( 1 + (0.939 + 0.342i)T \)
13 \( 1 + (-0.939 + 0.342i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (0.939 + 0.342i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.766 - 0.642i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (0.939 + 0.342i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.173 + 0.984i)T \)
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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

−26.82747518423557906752367719678, −25.86563523166363900675361259701, −24.98081550350702454907285022192, −24.472781121095194564233253854817, −23.17906041692637593800485151802, −22.19906901711574886071452373091, −21.6503813240939081341214852541, −20.08078482315823049754143801625, −18.95407625794437758056867813511, −18.02592153262231014023790543323, −17.16249565686428194890909840293, −16.46908877444483464267408058339, −15.07812242816173353760713677384, −14.3312716589226588108184970440, −13.54022377169846493317510797028, −12.32066686214564600993250545142, −10.69690776535626044655058280152, −9.633023541019582989557695182966, −8.89074801540578885481516233727, −7.44691120481964312309004514588, −6.52226119687942681874781646791, −5.5614900776186235476252685295, −4.380389300921657566491584149559, −2.60807287577736824430761153546, −0.75424652631817259374242435796, 1.26823899633766949008810907680, 2.2000857214680231915792746947, 3.75013085204965185773822332538, 4.920959607491271147690026750834, 6.18701456669207184268724910648, 7.82691068742259815401064407305, 9.14936605769861222938568634183, 9.756219003248618940771762471, 10.78756118897612701318013451935, 12.16507789396617608156653828085, 12.70583065433417551774612236526, 14.01475289115560281398391963847, 14.6528689884774682048220447108, 16.654467339324490545970954114920, 17.25478288919535137819203804206, 18.15668450754641949801694673169, 19.38815168730089604493247545023, 20.01823689388653984635184223605, 21.31385211498854371859233072618, 21.67386726929879499540757241676, 22.75037581645466127721378714257, 23.89823125493227838955438676795, 25.16785817538909714851480119812, 25.88734801473716372574112239270, 27.160269992021269347968178928659

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