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Properties

Label 2-189-9.4-c3-0-4
Degree $2$
Conductor $189$
Sign $-0.745 + 0.666i$
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  + (2.62 + 4.54i)2-s + (−9.74 + 16.8i)4-s + (−2.03 + 3.51i)5-s + (3.5 + 6.06i)7-s − 60.2·8-s − 21.2·10-s + (−7.57 − 13.1i)11-s + (−24.2 + 42.0i)13-s + (−18.3 + 31.7i)14-s + (−80.0 − 138. i)16-s − 107.·17-s + 109.·19-s + (−39.5 − 68.5i)20-s + (39.7 − 68.8i)22-s + (92.3 − 159. i)23-s + ⋯
L(s)  = 1  + (0.926 + 1.60i)2-s + (−1.21 + 2.11i)4-s + (−0.181 + 0.314i)5-s + (0.188 + 0.327i)7-s − 2.66·8-s − 0.673·10-s + (−0.207 − 0.359i)11-s + (−0.518 + 0.897i)13-s + (−0.350 + 0.606i)14-s + (−1.25 − 2.16i)16-s − 1.53·17-s + 1.32·19-s + (−0.442 − 0.766i)20-s + (0.384 − 0.666i)22-s + (0.837 − 1.44i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.745 + 0.666i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.745 + 0.666i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.690527 - 1.81011i\)
\(L(\frac12)\) \(\approx\) \(0.690527 - 1.81011i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.5 - 6.06i)T \)
good2 \( 1 + (-2.62 - 4.54i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (2.03 - 3.51i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (7.57 + 13.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (24.2 - 42.0i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 107.T + 4.91e3T^{2} \)
19 \( 1 - 109.T + 6.85e3T^{2} \)
23 \( 1 + (-92.3 + 159. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-47.0 - 81.4i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (67.6 - 117. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 149.T + 5.06e4T^{2} \)
41 \( 1 + (148. - 258. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-193. - 334. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (36.7 + 63.6i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 633.T + 1.48e5T^{2} \)
59 \( 1 + (162. - 281. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-34.7 - 60.2i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-139. + 242. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 497.T + 3.57e5T^{2} \)
73 \( 1 - 457.T + 3.89e5T^{2} \)
79 \( 1 + (-548. - 949. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-39.3 - 68.1i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 292.T + 7.04e5T^{2} \)
97 \( 1 + (82.2 + 142. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.04843426905374747523210733018, −12.07917230417496472483589678157, −11.00652650660538849658018273919, −9.188520839142972476863580298416, −8.439041856234090756459459518025, −7.15752287728600056225936087117, −6.63557991126300018954369516796, −5.28784945896513681885268158694, −4.45890185756917765792410508055, −2.98111015325823088279034742187, 0.63504101677335192491793836038, 2.21605179294778495611864595173, 3.51916247805068360018953952556, 4.69943363266976151062684382198, 5.51263066634268295138994185775, 7.31454989437977748150013560717, 8.914527529565292412709594883862, 9.915395684358842891374980485889, 10.75908176256440248380104531341, 11.62810517231761607484127312636

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