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Properties

Label 2-189-63.41-c3-0-4
Degree $2$
Conductor $189$
Sign $-0.872 + 0.487i$
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 + (4.49 + 17.9i)7-s − 11.2i·8-s − 8.03i·10-s + (−0.219 + 0.126i)11-s + (12.3 + 7.15i)13-s + (−9.34 − 9.66i)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 + (0.242 + 0.970i)7-s − 0.496i·8-s − 0.253i·10-s + (−0.00602 + 0.00347i)11-s + (0.264 + 0.152i)13-s + (−0.178 − 0.184i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.487i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.872 + 0.487i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.872 + 0.487i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.122648 - 0.470913i\)
\(L(\frac12)\) \(\approx\) \(0.122648 - 0.470913i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.49 - 17.9i)T \)
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

−12.81638031027982385501672346337, −11.35822639867441244604124103500, −11.17711017911186389243816335277, −9.257619358935915969480186879907, −8.844422124054812420749190146016, −7.51967807351467899672277516008, −6.78449425459107898078676266093, −5.14254503712661017025822719538, −3.75943441318456155534338052310, −2.54617390320235128047783283866, 0.23875120970207516901030231919, 1.54935160705173674070297311897, 3.99541372789805991505912629686, 4.84108075958906393852635992465, 6.13806598796080207971279410412, 7.62181220840458404204076893710, 8.609901678682180082199247866780, 9.489912236335785395974940668314, 10.63566257987715933154839665316, 11.25800123314708834721607958276

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