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Properties

Label 2-189-9.4-c3-0-7
Degree $2$
Conductor $189$
Sign $0.424 + 0.905i$
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  + (−1.96 − 3.40i)2-s + (−3.73 + 6.47i)4-s + (−1.21 + 2.10i)5-s + (3.5 + 6.06i)7-s − 2.05·8-s + 9.56·10-s + (21.0 + 36.5i)11-s + (19.7 − 34.2i)13-s + (13.7 − 23.8i)14-s + (33.9 + 58.8i)16-s + 2.07·17-s + 96.1·19-s + (−9.09 − 15.7i)20-s + (82.9 − 143. i)22-s + (36.8 − 63.8i)23-s + ⋯
L(s)  = 1  + (−0.695 − 1.20i)2-s + (−0.467 + 0.809i)4-s + (−0.108 + 0.188i)5-s + (0.188 + 0.327i)7-s − 0.0908·8-s + 0.302·10-s + (0.577 + 1.00i)11-s + (0.422 − 0.731i)13-s + (0.262 − 0.455i)14-s + (0.530 + 0.918i)16-s + 0.0296·17-s + 1.16·19-s + (−0.101 − 0.176i)20-s + (0.803 − 1.39i)22-s + (0.334 − 0.579i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.424 + 0.905i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.424 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.424 + 0.905i$
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: \((2,\ 189,\ (\ :3/2),\ 0.424 + 0.905i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.01125 - 0.642941i\)
\(L(\frac12)\) \(\approx\) \(1.01125 - 0.642941i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.96 + 3.40i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (1.21 - 2.10i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-21.0 - 36.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-19.7 + 34.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 2.07T + 4.91e3T^{2} \)
19 \( 1 - 96.1T + 6.85e3T^{2} \)
23 \( 1 + (-36.8 + 63.8i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-9.60 - 16.6i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-119. + 207. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 144.T + 5.06e4T^{2} \)
41 \( 1 + (36.1 - 62.5i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (240. + 416. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-147. - 255. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 627.T + 1.48e5T^{2} \)
59 \( 1 + (-74.8 + 129. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-315. - 546. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (2.02 - 3.50i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 798.T + 3.57e5T^{2} \)
73 \( 1 - 444.T + 3.89e5T^{2} \)
79 \( 1 + (-287. - 498. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-645. - 1.11e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 750.T + 7.04e5T^{2} \)
97 \( 1 + (209. + 363. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.81733163615114882137079096304, −10.92896588055185521502048704702, −10.01384568173020118796745547120, −9.225770476253265682674251695141, −8.206191145665391804913192333169, −6.90987556553866616408430285217, −5.40033297854929968563922851941, −3.73775264881514235088922960795, −2.46436668153094034973299523077, −1.03908850592185535062606059835, 0.938884728627878963625277599105, 3.43705764184748132331434322219, 5.10245191537661444916821387885, 6.31644694859926315225377996568, 7.12603194911433359822943025419, 8.292394400299045961014825209396, 8.906184973674220487632993384770, 9.969083947170120869723631508837, 11.31136269539360432690470090499, 12.12863558849309660777138956433

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