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Properties

Label 2-189-63.38-c3-0-3
Degree $2$
Conductor $189$
Sign $-0.684 + 0.729i$
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  + 3.72i·2-s − 5.84·4-s + (−1.33 + 2.30i)5-s + (8.98 − 16.1i)7-s + 8.01i·8-s + (−8.57 − 4.94i)10-s + (−58.8 + 33.9i)11-s + (−69.6 + 40.2i)13-s + (60.2 + 33.4i)14-s − 76.5·16-s + (29.6 − 51.2i)17-s + (−5.86 + 3.38i)19-s + (7.77 − 13.4i)20-s + (−126. − 219. i)22-s + (−109. − 63.2i)23-s + ⋯
L(s)  = 1  + 1.31i·2-s − 0.730·4-s + (−0.118 + 0.206i)5-s + (0.485 − 0.874i)7-s + 0.354i·8-s + (−0.271 − 0.156i)10-s + (−1.61 + 0.931i)11-s + (−1.48 + 0.858i)13-s + (1.15 + 0.638i)14-s − 1.19·16-s + (0.422 − 0.731i)17-s + (−0.0707 + 0.0408i)19-s + (0.0869 − 0.150i)20-s + (−1.22 − 2.12i)22-s + (−0.992 − 0.572i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.684 + 0.729i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.684 + 0.729i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.288871 - 0.667402i\)
\(L(\frac12)\) \(\approx\) \(0.288871 - 0.667402i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-8.98 + 16.1i)T \)
good2 \( 1 - 3.72iT - 8T^{2} \)
5 \( 1 + (1.33 - 2.30i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (58.8 - 33.9i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (69.6 - 40.2i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-29.6 + 51.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (5.86 - 3.38i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (109. + 63.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-114. - 66.0i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 72.5iT - 2.97e4T^{2} \)
37 \( 1 + (63.8 + 110. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-4.93 - 8.54i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (108. - 188. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 185.T + 1.03e5T^{2} \)
53 \( 1 + (174. + 100. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 178.T + 2.05e5T^{2} \)
61 \( 1 - 225. iT - 2.26e5T^{2} \)
67 \( 1 - 438.T + 3.00e5T^{2} \)
71 \( 1 - 533. iT - 3.57e5T^{2} \)
73 \( 1 + (-287. - 165. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 1.00e3T + 4.93e5T^{2} \)
83 \( 1 + (590. - 1.02e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-66.3 - 114. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-761. - 439. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.91771798954306074585332406112, −11.79786469210714069873464962833, −10.59587481867916970129825135455, −9.703496447053793983995153552591, −8.213901711978636965568974737437, −7.36859621161865560913388604707, −6.94487646553114203895491299979, −5.24034321558652689110722690924, −4.59717228540215416157258042686, −2.39166611169688723026605548445, 0.29398703964626688357686581060, 2.20513767161883605247541153677, 3.09008689847347814934849598347, 4.78454399383585802695636860415, 5.83027298951124244959142164501, 7.73532595163567828864256112210, 8.528120931668056561927307227653, 9.969995576231228407613470734627, 10.48115944232414676855736181406, 11.58918950280620264499788032664

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