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Properties

Label 2-189-7.2-c3-0-17
Degree $2$
Conductor $189$
Sign $0.972 - 0.234i$
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  + (−2.38 + 4.12i)2-s + (−7.33 − 12.7i)4-s + (−9.35 + 16.1i)5-s + (14.2 − 11.8i)7-s + 31.8·8-s + (−44.5 − 77.1i)10-s + (−29.7 − 51.4i)11-s − 12.2·13-s + (15.1 + 86.8i)14-s + (−17.0 + 29.4i)16-s + (−5.76 − 9.97i)17-s + (−24.9 + 43.1i)19-s + 274.·20-s + 283.·22-s + (62.8 − 108. i)23-s + ⋯
L(s)  = 1  + (−0.841 + 1.45i)2-s + (−0.917 − 1.58i)4-s + (−0.836 + 1.44i)5-s + (0.767 − 0.641i)7-s + 1.40·8-s + (−1.40 − 2.43i)10-s + (−0.814 − 1.41i)11-s − 0.260·13-s + (0.289 + 1.65i)14-s + (−0.265 + 0.460i)16-s + (−0.0821 − 0.142i)17-s + (−0.300 + 0.520i)19-s + 3.06·20-s + 2.74·22-s + (0.570 − 0.987i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.972 - 0.234i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.972 - 0.234i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.535755 + 0.0635795i\)
\(L(\frac12)\) \(\approx\) \(0.535755 + 0.0635795i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-14.2 + 11.8i)T \)
good2 \( 1 + (2.38 - 4.12i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (9.35 - 16.1i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (29.7 + 51.4i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 12.2T + 2.19e3T^{2} \)
17 \( 1 + (5.76 + 9.97i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (24.9 - 43.1i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-62.8 + 108. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 228.T + 2.43e4T^{2} \)
31 \( 1 + (-94.7 - 164. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (16.5 - 28.7i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 524.T + 6.89e4T^{2} \)
43 \( 1 - 234.T + 7.95e4T^{2} \)
47 \( 1 + (-136. + 237. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (127. + 221. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (84.4 + 146. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-97.5 + 168. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (257. + 446. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 319.T + 3.57e5T^{2} \)
73 \( 1 + (317. + 550. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-426. + 737. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 264.T + 5.71e5T^{2} \)
89 \( 1 + (-457. + 791. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 455.T + 9.12e5T^{2} \)
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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

−11.85764169746453579793117541403, −10.66008509520163966763561025947, −10.37410328598890444273678540529, −8.507711338523717572048444223253, −8.044001704165547305186488762652, −7.07168240971046728719434960148, −6.34913660324455426958195865379, −4.90141405043495754226592101141, −3.16220010009531246388085761896, −0.36653202769270927681933868269, 1.19507446400328842937925994278, 2.47493151837956601806302794081, 4.29746553377812740709549112072, 5.07776213512682416163572887746, 7.60131774498690396179805214445, 8.374428640019347465456452249464, 9.132543974382755802662318424878, 10.07074679045479881539102014966, 11.24693320670175339103355874324, 12.05241024576072855203139251832

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