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Properties

Label 2-189-7.4-c3-0-31
Degree $2$
Conductor $189$
Sign $-0.917 + 0.396i$
Analytic cond. $11.1513$
Root an. cond. $3.33936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.884 + 1.53i)2-s + (2.43 − 4.21i)4-s + (−6.52 − 11.3i)5-s + (−18.4 − 0.966i)7-s + 22.7·8-s + (11.5 − 19.9i)10-s + (−26.4 + 45.8i)11-s − 71.7·13-s + (−14.8 − 29.1i)14-s + (0.663 + 1.14i)16-s + (−53.2 + 92.2i)17-s + (−26.9 − 46.7i)19-s − 63.5·20-s − 93.6·22-s + (−9.32 − 16.1i)23-s + ⋯
L(s)  = 1  + (0.312 + 0.541i)2-s + (0.304 − 0.527i)4-s + (−0.583 − 1.01i)5-s + (−0.998 − 0.0522i)7-s + 1.00·8-s + (0.365 − 0.632i)10-s + (−0.725 + 1.25i)11-s − 1.52·13-s + (−0.284 − 0.557i)14-s + (0.0103 + 0.0179i)16-s + (−0.760 + 1.31i)17-s + (−0.325 − 0.563i)19-s − 0.710·20-s − 0.907·22-s + (−0.0845 − 0.146i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.396i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.917 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.917 + 0.396i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.917 + 0.396i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0766041 - 0.370287i\)
\(L(\frac12)\) \(\approx\) \(0.0766041 - 0.370287i\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.4 + 0.966i)T \)
good2 \( 1 + (-0.884 - 1.53i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (6.52 + 11.3i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (26.4 - 45.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 71.7T + 2.19e3T^{2} \)
17 \( 1 + (53.2 - 92.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (26.9 + 46.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (9.32 + 16.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 261.T + 2.43e4T^{2} \)
31 \( 1 + (-61.1 + 105. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (139. + 240. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 31.3T + 6.89e4T^{2} \)
43 \( 1 + 347.T + 7.95e4T^{2} \)
47 \( 1 + (271. + 469. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-128. + 222. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (157. - 273. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (69.4 + 120. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (198. - 344. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 843.T + 3.57e5T^{2} \)
73 \( 1 + (-436. + 755. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-277. - 480. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 297.T + 5.71e5T^{2} \)
89 \( 1 + (51.2 + 88.7i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 515.T + 9.12e5T^{2} \)
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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

−12.03117790526865345270574285450, −10.43021214573361830256533125756, −9.855670987842234339027930140182, −8.512020916926113184814723248941, −7.33409423546335944296319309085, −6.50814090720511389805486470348, −5.06555736921552801193022990457, −4.38113608137037837158208576663, −2.21870132029935207926582761390, −0.13653547828827663157199457907, 2.81962157662477851083733532657, 3.11443697521656853295142199567, 4.74738847110645436140468926181, 6.53445561401830796139607866521, 7.28775059322326560208887257976, 8.358802285204833381904426191230, 9.928490817020079564447301034068, 10.72015124114656030898045870459, 11.65629299538665751281542355934, 12.33014701446284728240424902788

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