Properties

Label 2-189-21.17-c1-0-1
Degree $2$
Conductor $189$
Sign $0.205 - 0.978i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)4-s + (0.5 + 2.59i)7-s + 6.92i·13-s + (−1.99 − 3.46i)16-s + (4.5 − 2.59i)19-s + (2.5 − 4.33i)25-s + (−5 − 1.73i)28-s + (−1.5 − 0.866i)31-s + (−5 − 8.66i)37-s + 13·43-s + (−6.5 + 2.59i)49-s + (−11.9 − 6.92i)52-s + (−7.5 + 4.33i)61-s + 7.99·64-s + (8 − 13.8i)67-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + (0.188 + 0.981i)7-s + 1.92i·13-s + (−0.499 − 0.866i)16-s + (1.03 − 0.596i)19-s + (0.5 − 0.866i)25-s + (−0.944 − 0.327i)28-s + (−0.269 − 0.155i)31-s + (−0.821 − 1.42i)37-s + 1.98·43-s + (−0.928 + 0.371i)49-s + (−1.66 − 0.960i)52-s + (−0.960 + 0.554i)61-s + 0.999·64-s + (0.977 − 1.69i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.205 - 0.978i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.205 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.802283 + 0.651483i\)
\(L(\frac12)\) \(\approx\) \(0.802283 + 0.651483i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-0.5 - 2.59i)T \)
good2 \( 1 + (1 - 1.73i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.92iT - 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.5 + 2.59i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 13T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.5 - 4.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8 + 13.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-13.5 - 7.79i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 19.0iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.55212682660438983472157645227, −11.97555252763756146134648577293, −11.08995198316239262165038813214, −9.288465943745865816007739079544, −9.036360803211178457930941565742, −7.76823041981533334697404465071, −6.65677308889071393553363619996, −5.14979491245933178669563729918, −4.00740754937879545210542399250, −2.43514023894947911090584523261, 1.03997947285202387552651717274, 3.42704674854982646766031179707, 4.91529950925449047407897129273, 5.80736724567364436227539884582, 7.27141302921257274556978944484, 8.270771669586352006171892199710, 9.602976770644561260497905342236, 10.38152969226145926335818366762, 11.05401100186968864581132446171, 12.54713968965605036693842573810

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