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Properties

Label 1-189-189.115-r1-0-0
Degree $1$
Conductor $189$
Sign $0.240 - 0.970i$
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 + (−0.173 + 0.984i)5-s + (−0.5 − 0.866i)8-s − 10-s + (0.173 + 0.984i)11-s + (−0.766 − 0.642i)13-s + (0.766 − 0.642i)16-s − 17-s − 19-s + (−0.173 − 0.984i)20-s + (−0.939 + 0.342i)22-s + (0.766 + 0.642i)23-s + (−0.939 − 0.342i)25-s + (0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)4-s + (−0.173 + 0.984i)5-s + (−0.5 − 0.866i)8-s − 10-s + (0.173 + 0.984i)11-s + (−0.766 − 0.642i)13-s + (0.766 − 0.642i)16-s − 17-s − 19-s + (−0.173 − 0.984i)20-s + (−0.939 + 0.342i)22-s + (0.766 + 0.642i)23-s + (−0.939 − 0.342i)25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.240 - 0.970i$
Analytic conductor: \(20.3108\)
Root analytic conductor: \(20.3108\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (1:\ ),\ 0.240 - 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07634680970 + 0.05972495586i\)
\(L(\frac12)\) \(\approx\) \(-0.07634680970 + 0.05972495586i\)
\(L(1)\) \(\approx\) \(0.5885553111 + 0.4915736320i\)
\(L(1)\) \(\approx\) \(0.5885553111 + 0.4915736320i\)

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 \( 1 + (-0.173 + 0.984i)T \)
11 \( 1 + (0.173 + 0.984i)T \)
13 \( 1 + (-0.766 - 0.642i)T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (0.766 - 0.642i)T \)
31 \( 1 + (0.939 - 0.342i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
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 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (-0.766 + 0.642i)T \)
89 \( 1 - T \)
97 \( 1 + (0.939 + 0.342i)T \)
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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

−26.62303863992572071013970703770, −24.95365461270985249830015421523, −24.064148064302195366183931147891, −23.31387601022232323559271495434, −21.96770545332132342397273075008, −21.39433465401169780765216457408, −20.35451569280338061532925625337, −19.51916921272632407780729465727, −18.79252375314644774921065797349, −17.38624555652959975679936578561, −16.64898663688813114838455596224, −15.26223548829367492268941888699, −14.01588738353383255735212946629, −13.14619324164657551732064100131, −12.20966308597060631663182775492, −11.35029314927550089060276037584, −10.221493507020237773232203406978, −8.92363663187002065296003554641, −8.44845056849067369065396183508, −6.49302625263599910303025578640, −4.998419631093380057908271197497, −4.274865103126844346133258287600, −2.80300470364335532973003019017, −1.38069364678973294077435911943, −0.03283119134821446874866769126, 2.47323860356742549192064780658, 3.927630646380995910796005111022, 5.04135254834188838651656976832, 6.475453734699807963061995746161, 7.14550306503204413850265966410, 8.20688962682635337289263769193, 9.5466735291287759399153286563, 10.54210703642952709112961881138, 12.005194737552851727662157913420, 13.08818955397413326754712738755, 14.182918177396617423789273210781, 15.21009820127562464208290756367, 15.49901518695791665250190062700, 17.26985703633557415639188477352, 17.59564500087733305781420544318, 18.82723052204111227251891439, 19.738123781069040728679263617067, 21.27476228925642774791281259604, 22.31792596267993455236014246864, 22.86945820290641252275703927632, 23.77534473496867418680981118065, 25.00184805937366887034736421899, 25.582582288088921549552524884661, 26.66866654291733589895261832136, 27.2248084744359650578049300431

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