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Properties

Label 2-189-63.58-c3-0-0
Degree $2$
Conductor $189$
Sign $-0.905 + 0.424i$
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  − 5.43·2-s + 21.4·4-s + (6.55 + 11.3i)5-s + (−18.0 + 3.92i)7-s − 73.2·8-s + (−35.6 − 61.6i)10-s + (−9.12 + 15.8i)11-s + (−12.9 + 22.3i)13-s + (98.2 − 21.3i)14-s + 225.·16-s + (−1.04 − 1.80i)17-s + (10.2 − 17.8i)19-s + (140. + 244. i)20-s + (49.5 − 85.8i)22-s + (34.2 + 59.2i)23-s + ⋯
L(s)  = 1  − 1.91·2-s + 2.68·4-s + (0.586 + 1.01i)5-s + (−0.977 + 0.212i)7-s − 3.23·8-s + (−1.12 − 1.95i)10-s + (−0.250 + 0.433i)11-s + (−0.275 + 0.477i)13-s + (1.87 − 0.407i)14-s + 3.52·16-s + (−0.0148 − 0.0257i)17-s + (0.124 − 0.215i)19-s + (1.57 + 2.72i)20-s + (0.480 − 0.832i)22-s + (0.310 + 0.537i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.905 + 0.424i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.905 + 0.424i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0235914 - 0.105965i\)
\(L(\frac12)\) \(\approx\) \(0.0235914 - 0.105965i\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.0 - 3.92i)T \)
good2 \( 1 + 5.43T + 8T^{2} \)
5 \( 1 + (-6.55 - 11.3i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (9.12 - 15.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (12.9 - 22.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (1.04 + 1.80i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-10.2 + 17.8i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-34.2 - 59.2i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (132. + 229. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 30.4T + 2.97e4T^{2} \)
37 \( 1 + (143. - 249. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-152. + 263. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (119. + 207. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 419.T + 1.03e5T^{2} \)
53 \( 1 + (-114. - 198. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 799.T + 2.05e5T^{2} \)
61 \( 1 - 94.2T + 2.26e5T^{2} \)
67 \( 1 + 569.T + 3.00e5T^{2} \)
71 \( 1 - 211.T + 3.57e5T^{2} \)
73 \( 1 + (314. + 544. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 334.T + 4.93e5T^{2} \)
83 \( 1 + (163. + 282. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (160. - 278. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (60.9 + 105. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.22679558766513382810389142391, −11.26666206524005252885250784199, −10.28292919904778708170778858607, −9.743855450651148741932873635887, −8.958891296552018156583787697832, −7.56223802270483307454884769153, −6.81615958365179999343535844087, −5.97565769584114977691590861922, −3.05411024935447359663827285787, −2.00131407337185251660440342516, 0.087542886080611562461604357506, 1.40392597699866236931226445026, 3.02380603729925844106495610133, 5.51955560324040772346977018077, 6.62954364068923796084417154103, 7.73478764419327629536565828099, 8.805518454362179686266708934395, 9.398198625644219665454859263927, 10.23433343800956068889042992582, 11.10006487697557449204152159113

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