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Properties

Label 2-189-63.25-c3-0-13
Degree $2$
Conductor $189$
Sign $-0.109 + 0.993i$
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  + 0.438·2-s − 7.80·4-s + (−8.04 + 13.9i)5-s + (16.9 − 7.49i)7-s − 6.93·8-s + (−3.52 + 6.11i)10-s + (−13.2 − 22.9i)11-s + (−7.29 − 12.6i)13-s + (7.42 − 3.28i)14-s + 59.4·16-s + (17.3 − 29.9i)17-s + (−65.3 − 113. i)19-s + (62.8 − 108. i)20-s + (−5.82 − 10.0i)22-s + (33.0 − 57.1i)23-s + ⋯
L(s)  = 1  + 0.155·2-s − 0.975·4-s + (−0.719 + 1.24i)5-s + (0.914 − 0.404i)7-s − 0.306·8-s + (−0.111 + 0.193i)10-s + (−0.363 − 0.630i)11-s + (−0.155 − 0.269i)13-s + (0.141 − 0.0627i)14-s + 0.928·16-s + (0.247 − 0.427i)17-s + (−0.789 − 1.36i)19-s + (0.702 − 1.21i)20-s + (−0.0564 − 0.0977i)22-s + (0.299 − 0.518i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.109 + 0.993i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.109 + 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.486817 - 0.543604i\)
\(L(\frac12)\) \(\approx\) \(0.486817 - 0.543604i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-16.9 + 7.49i)T \)
good2 \( 1 - 0.438T + 8T^{2} \)
5 \( 1 + (8.04 - 13.9i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (13.2 + 22.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (7.29 + 12.6i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-17.3 + 29.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (65.3 + 113. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-33.0 + 57.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-102. + 177. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 253.T + 2.97e4T^{2} \)
37 \( 1 + (-111. - 193. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (42.4 + 73.6i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (16.6 - 28.8i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 525.T + 1.03e5T^{2} \)
53 \( 1 + (-196. + 339. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 649.T + 2.05e5T^{2} \)
61 \( 1 - 38.7T + 2.26e5T^{2} \)
67 \( 1 + 180.T + 3.00e5T^{2} \)
71 \( 1 + 624.T + 3.57e5T^{2} \)
73 \( 1 + (-155. + 270. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + 487.T + 4.93e5T^{2} \)
83 \( 1 + (345. - 598. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-525. - 909. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (86.6 - 149. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.61212722124597176706304998392, −10.97094664432991940160617288565, −10.03868812541546159529578042181, −8.621053599700457433711947641614, −7.82784316190882251659482726485, −6.74225015473297536840187300520, −5.18854736310427299982018929987, −4.11160124754608067606612655357, −2.87211044540419096839941966819, −0.33565212010878247057420382807, 1.48738944949376311894922639983, 3.89529328711534249586901906675, 4.77962444313895731126823074493, 5.57522399103338795752737048536, 7.63734502332526657438836263923, 8.428928252014569905597370743848, 9.091082290833822530661500684449, 10.32631954098467254719878066840, 11.69444048842286226001990670176, 12.56121820285507963340993311906

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