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Properties

Label 2-189-63.4-c3-0-21
Degree $2$
Conductor $189$
Sign $-0.396 - 0.918i$
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  + (2.71 − 4.70i)2-s + (−10.7 − 18.6i)4-s − 13.1·5-s + (12.4 − 13.7i)7-s − 73.2·8-s + (−35.6 + 61.6i)10-s + 18.2·11-s + (−12.9 + 22.3i)13-s + (−30.6 − 95.7i)14-s + (−112. + 195. i)16-s + (−1.04 + 1.80i)17-s + (10.2 + 17.8i)19-s + (140. + 244. i)20-s + (49.5 − 85.8i)22-s − 68.4·23-s + ⋯
L(s)  = 1  + (0.959 − 1.66i)2-s + (−1.34 − 2.32i)4-s − 1.17·5-s + (0.672 − 0.740i)7-s − 3.23·8-s + (−1.12 + 1.95i)10-s + 0.500·11-s + (−0.275 + 0.477i)13-s + (−0.585 − 1.82i)14-s + (−1.76 + 3.05i)16-s + (−0.0148 + 0.0257i)17-s + (0.124 + 0.215i)19-s + (1.57 + 2.72i)20-s + (0.480 − 0.832i)22-s − 0.620·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.396 - 0.918i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.396 - 0.918i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.811449 + 1.23398i\)
\(L(\frac12)\) \(\approx\) \(0.811449 + 1.23398i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-12.4 + 13.7i)T \)
good2 \( 1 + (-2.71 + 4.70i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 13.1T + 125T^{2} \)
11 \( 1 - 18.2T + 1.33e3T^{2} \)
13 \( 1 + (12.9 - 22.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (1.04 - 1.80i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-10.2 - 17.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 68.4T + 1.21e4T^{2} \)
29 \( 1 + (132. + 229. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-15.2 - 26.3i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (143. + 249. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-152. + 263. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (119. + 207. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-209. + 363. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-114. + 198. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-399. - 692. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (47.1 - 81.6i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-284. - 492. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 211.T + 3.57e5T^{2} \)
73 \( 1 + (314. - 544. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-167. + 289. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (163. + 282. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (160. + 278. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (60.9 + 105. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.74553076311088031463617327331, −10.81997013939667813076154696890, −9.929418986083961318212671019346, −8.670076507535869383482907884465, −7.26786601208469065626768268505, −5.53001205072932109905438833705, −4.12841792067314901635337141541, −3.87829522093895843391708754845, −2.03923876886209294909489696507, −0.48355521711620227241055045054, 3.28633492209112590968477788814, 4.44408967884547089531613410173, 5.34468432676611701805472083789, 6.53500531053340036587609399831, 7.66212296258542657253159917917, 8.200115103428468693292959955544, 9.216124566087413709204438110202, 11.32999817416762119351613966890, 12.13753273495115734919066585397, 12.84837475934290767401517429734

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