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Properties

Label 2-189-189.184-c1-0-13
Degree $2$
Conductor $189$
Sign $0.997 + 0.0696i$
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  + (0.337 + 1.91i)2-s + (−1.21 − 1.23i)3-s + (−1.67 + 0.610i)4-s + (0.582 − 3.30i)5-s + (1.95 − 2.74i)6-s + (0.666 − 2.56i)7-s + (0.209 + 0.362i)8-s + (−0.0477 + 2.99i)9-s + 6.52·10-s + (−0.778 − 4.41i)11-s + (2.79 + 1.32i)12-s + (2.58 + 2.16i)13-s + (5.13 + 0.411i)14-s + (−4.78 + 3.29i)15-s + (−3.35 + 2.81i)16-s − 3.96·17-s + ⋯
L(s)  = 1  + (0.238 + 1.35i)2-s + (−0.701 − 0.712i)3-s + (−0.838 + 0.305i)4-s + (0.260 − 1.47i)5-s + (0.798 − 1.12i)6-s + (0.251 − 0.967i)7-s + (0.0739 + 0.128i)8-s + (−0.0159 + 0.999i)9-s + 2.06·10-s + (−0.234 − 1.33i)11-s + (0.805 + 0.383i)12-s + (0.715 + 0.600i)13-s + (1.37 + 0.110i)14-s + (−1.23 + 0.850i)15-s + (−0.839 + 0.704i)16-s − 0.961·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.997 + 0.0696i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.997 + 0.0696i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13640 - 0.0396242i\)
\(L(\frac12)\) \(\approx\) \(1.13640 - 0.0396242i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.21 + 1.23i)T \)
7 \( 1 + (-0.666 + 2.56i)T \)
good2 \( 1 + (-0.337 - 1.91i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (-0.582 + 3.30i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (0.778 + 4.41i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-2.58 - 2.16i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + 3.96T + 17T^{2} \)
19 \( 1 - 4.97T + 19T^{2} \)
23 \( 1 + (-1.70 - 1.42i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (3.26 - 2.74i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-2.19 + 0.797i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-1.49 - 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.219 - 0.184i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-9.46 - 3.44i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (5.08 + 1.85i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-5.11 - 8.86i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.56 - 2.14i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (9.73 + 3.54i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (2.20 - 12.5i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-2.77 + 4.81i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.30 - 3.98i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.215 - 1.22i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (8.72 - 7.31i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 + (-10.3 - 3.77i)T + (74.3 + 62.3i)T^{2} \)
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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

−13.09097141812424464278152606594, −11.59260834868529322855721819448, −10.88179500939362253454272678548, −9.052308974110756677510573769866, −8.215744349218581817845644255521, −7.30850321959905503880030859372, −6.18773898028186339500780672727, −5.37616174663473744856069146855, −4.41906898719021700255918253340, −1.17200292352205556248723682028, 2.28106802109118743498879268721, 3.38205325759317146014902551432, 4.75587306544414890776834445605, 6.06811917333869690213188734565, 7.24402723957042379554295131980, 9.230012502763309637279286055862, 10.03983409956380699724204510524, 10.80028363510159755159972062822, 11.40103036961081127429843845357, 12.21666399168093244168114015658

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