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Properties

Label 2-189-21.5-c1-0-0
Degree $2$
Conductor $189$
Sign $-0.575 - 0.817i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.568 − 0.328i)2-s + (−0.784 − 1.35i)4-s + (−1.65 + 2.86i)5-s + (−2.58 + 0.568i)7-s + 2.34i·8-s + (1.88 − 1.08i)10-s + (−2.02 + 1.17i)11-s + 1.58i·13-s + (1.65 + 0.524i)14-s + (−0.799 + 1.38i)16-s + (−0.568 − 0.984i)17-s + (3.85 + 2.22i)19-s + 5.19·20-s + 1.53·22-s + (−8.13 − 4.69i)23-s + ⋯
L(s)  = 1  + (−0.402 − 0.232i)2-s + (−0.392 − 0.679i)4-s + (−0.740 + 1.28i)5-s + (−0.976 + 0.214i)7-s + 0.828i·8-s + (0.595 − 0.343i)10-s + (−0.611 + 0.353i)11-s + 0.438i·13-s + (0.442 + 0.140i)14-s + (−0.199 + 0.346i)16-s + (−0.137 − 0.238i)17-s + (0.884 + 0.510i)19-s + 1.16·20-s + 0.328·22-s + (−1.69 − 0.979i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.575 - 0.817i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.575 - 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.144805 + 0.279054i\)
\(L(\frac12)\) \(\approx\) \(0.144805 + 0.279054i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.58 - 0.568i)T \)
good2 \( 1 + (0.568 + 0.328i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.65 - 2.86i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.02 - 1.17i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.58iT - 13T^{2} \)
17 \( 1 + (0.568 + 0.984i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.85 - 2.22i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (8.13 + 4.69i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.65iT - 29T^{2} \)
31 \( 1 + (6.33 - 3.65i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.58 + 4.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.64T + 41T^{2} \)
43 \( 1 + 2.16T + 43T^{2} \)
47 \( 1 + (-2.79 + 4.83i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.70 - 5.02i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.08 + 1.88i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.46 - 2.00i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.38 - 9.32i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.39iT - 71T^{2} \)
73 \( 1 + (9.25 - 5.34i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.616 - 1.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.03T + 83T^{2} \)
89 \( 1 + (3.73 - 6.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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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

−12.79631040464534164699030426725, −11.73505633596881469249409224511, −10.71645953622773529739374402896, −10.09508801136472363260710020885, −9.148018542933677838575696420507, −7.80716832578018950694404244788, −6.76131985201308250489536087099, −5.64095016974115108971236841936, −3.95245326167322305843080169927, −2.52578972721512110181354071767, 0.31040621761631133695620562838, 3.37499441126844142504108111849, 4.42238938334291226059404401395, 5.86174813477627407485458287744, 7.54756649660085841446956074051, 8.056110752390070202067243996749, 9.171972566634210351257788698699, 9.850684891924086823223588325488, 11.44324220962201781179188089669, 12.45011230125114642536367056509

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