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Properties

Label 1-189-189.68-r0-0-0
Degree $1$
Conductor $189$
Sign $0.276 - 0.960i$
Analytic cond. $0.877712$
Root an. cond. $0.877712$
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  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.766 − 0.642i)5-s + (0.5 − 0.866i)8-s − 10-s + (−0.766 − 0.642i)11-s + (0.939 + 0.342i)13-s + (−0.939 + 0.342i)16-s + 17-s − 19-s + (0.766 + 0.642i)20-s + (0.173 + 0.984i)22-s + (0.939 + 0.342i)23-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.766 − 0.642i)5-s + (0.5 − 0.866i)8-s − 10-s + (−0.766 − 0.642i)11-s + (0.939 + 0.342i)13-s + (−0.939 + 0.342i)16-s + 17-s − 19-s + (0.766 + 0.642i)20-s + (0.173 + 0.984i)22-s + (0.939 + 0.342i)23-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.276 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.276 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.276 - 0.960i$
Analytic conductor: \(0.877712\)
Root analytic conductor: \(0.877712\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (0:\ ),\ 0.276 - 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7184588381 - 0.5406992755i\)
\(L(\frac12)\) \(\approx\) \(0.7184588381 - 0.5406992755i\)
\(L(1)\) \(\approx\) \(0.7826763225 - 0.3356930610i\)
\(L(1)\) \(\approx\) \(0.7826763225 - 0.3356930610i\)

Euler product

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

−27.218346776593100110928136327032, −26.14228539073910708424033694670, −25.544498517907289789092520340299, −24.90957956167195774552085579886, −23.3262669843642194059395326792, −23.0728243391375989350506883783, −21.4359790195269435481419215961, −20.62853337415337653726641053649, −19.29479243683887038555284539166, −18.40067906383850640246038217189, −17.76959952341568703815503053272, −16.78763282783901423323490077840, −15.65971247674474706363245703730, −14.77501746713812262359714500633, −13.86580412733227286366050730529, −12.655179647065207956544721460635, −10.82398354230088647129656331180, −10.3883380915130831994477638514, −9.22882631708407848008611323715, −8.10606945346922934361089342321, −6.97160911074909837947884412641, −6.043126222943600030683369004764, −4.97082395494321120620736258844, −2.90438269980137012660860118899, −1.507911024202578403190516997311, 1.04354239844577868844556489373, 2.34631270617018547234735534635, 3.710703066754069782172396244, 5.26380231680332926960702798020, 6.57746700585421917605864140190, 8.1470829924888248277886750406, 8.80887324204443514584925223887, 9.96339941509420571104808571413, 10.80281185624033774563094143129, 11.987546563542706060827398564315, 13.08927742294376607034776896415, 13.741340308279614006857305662930, 15.52969080979422523878737793705, 16.633651493014368445265817328461, 17.202767049027311504498619286773, 18.45094654743873201074943534789, 19.02349444442992400173261996570, 20.37752380429512020532298727129, 21.154345296206156918061862128692, 21.58475603987338956335483672400, 23.11382898662256662281769422812, 24.219751591867109540366118135059, 25.50253910315223421102799865809, 25.81853921390317802873192984964, 27.125586272684153239530226678508

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