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Properties

Label 2-189-63.20-c3-0-11
Degree $2$
Conductor $189$
Sign $0.483 + 0.875i$
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  + (−0.628 − 0.362i)2-s + (−3.73 − 6.47i)4-s + (5.53 + 9.58i)5-s + (13.3 − 12.8i)7-s + 11.2i·8-s − 8.03i·10-s + (−0.219 − 0.126i)11-s + (−12.3 + 7.15i)13-s + (−13.0 + 3.26i)14-s + (−25.8 + 44.7i)16-s + 92.5·17-s − 130. i·19-s + (41.3 − 71.6i)20-s + (0.0920 + 0.159i)22-s + (102. − 59.3i)23-s + ⋯
L(s)  = 1  + (−0.222 − 0.128i)2-s + (−0.467 − 0.808i)4-s + (0.494 + 0.857i)5-s + (0.718 − 0.695i)7-s + 0.496i·8-s − 0.253i·10-s + (−0.00602 − 0.00347i)11-s + (−0.264 + 0.152i)13-s + (−0.248 + 0.0622i)14-s + (−0.403 + 0.698i)16-s + 1.32·17-s − 1.57i·19-s + (0.462 − 0.800i)20-s + (0.000891 + 0.00154i)22-s + (0.931 − 0.538i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.483 + 0.875i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.483 + 0.875i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.34248 - 0.792354i\)
\(L(\frac12)\) \(\approx\) \(1.34248 - 0.792354i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-13.3 + 12.8i)T \)
good2 \( 1 + (0.628 + 0.362i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (-5.53 - 9.58i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (0.219 + 0.126i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (12.3 - 7.15i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 92.5T + 4.91e3T^{2} \)
19 \( 1 + 130. iT - 6.85e3T^{2} \)
23 \( 1 + (-102. + 59.3i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (248. + 143. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-214. + 123. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 188.T + 5.06e4T^{2} \)
41 \( 1 + (53.2 + 92.2i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (21.5 - 37.3i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-68.8 + 119. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 419. iT - 1.48e5T^{2} \)
59 \( 1 + (-217. - 376. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (163. + 94.3i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (185. + 321. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 26.1iT - 3.57e5T^{2} \)
73 \( 1 - 728. iT - 3.89e5T^{2} \)
79 \( 1 + (48.6 - 84.3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (401. - 695. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 236.T + 7.04e5T^{2} \)
97 \( 1 + (-1.26e3 - 732. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.57676648124991529051538304979, −10.80926295997396101383817543054, −10.06168250146723345110015165854, −9.188811115091243260986420175113, −7.83375490547440045540823533922, −6.72781371963665078639858234735, −5.51482704878716371358143809999, −4.38253438304422650683626067899, −2.50669714846664254856766192698, −0.872257839562017563224436858749, 1.41718726002485515239088500522, 3.32233747285408321072939782669, 4.85946348338082549474087032258, 5.69236508578667922233927494106, 7.49232115410974983259152317939, 8.295250614978760072721303336773, 9.133661365375616671383298828552, 9.997375314233894878523927008915, 11.55176939946496665814291533754, 12.43127005512440357024015925931

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