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Properties

Label 2-189-9.4-c3-0-8
Degree $2$
Conductor $189$
Sign $0.364 - 0.931i$
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  + (1.21 + 2.11i)2-s + (1.03 − 1.78i)4-s + (−2.61 + 4.53i)5-s + (−3.5 − 6.06i)7-s + 24.5·8-s − 12.7·10-s + (30.6 + 53.1i)11-s + (0.911 − 1.57i)13-s + (8.52 − 14.7i)14-s + (21.6 + 37.4i)16-s + 76.2·17-s + 75.2·19-s + (5.39 + 9.35i)20-s + (−74.7 + 129. i)22-s + (−65.7 + 113. i)23-s + ⋯
L(s)  = 1  + (0.430 + 0.746i)2-s + (0.128 − 0.223i)4-s + (−0.234 + 0.405i)5-s + (−0.188 − 0.327i)7-s + 1.08·8-s − 0.403·10-s + (0.840 + 1.45i)11-s + (0.0194 − 0.0336i)13-s + (0.162 − 0.281i)14-s + (0.337 + 0.585i)16-s + 1.08·17-s + 0.908·19-s + (0.0603 + 0.104i)20-s + (−0.724 + 1.25i)22-s + (−0.596 + 1.03i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.364 - 0.931i$
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: \((2,\ 189,\ (\ :3/2),\ 0.364 - 0.931i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.00537 + 1.36868i\)
\(L(\frac12)\) \(\approx\) \(2.00537 + 1.36868i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.21 - 2.11i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (2.61 - 4.53i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-30.6 - 53.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-0.911 + 1.57i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 76.2T + 4.91e3T^{2} \)
19 \( 1 - 75.2T + 6.85e3T^{2} \)
23 \( 1 + (65.7 - 113. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (32.0 + 55.4i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-88.7 + 153. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 337.T + 5.06e4T^{2} \)
41 \( 1 + (-22.8 + 39.5i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-146. - 254. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (163. + 282. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 66.4T + 1.48e5T^{2} \)
59 \( 1 + (344. - 597. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (336. + 582. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-288. + 499. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 4.08T + 3.57e5T^{2} \)
73 \( 1 - 747.T + 3.89e5T^{2} \)
79 \( 1 + (531. + 920. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (415. + 720. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 460.T + 7.04e5T^{2} \)
97 \( 1 + (431. + 747. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.31212119523073550319301367198, −11.41391265064521436103699150939, −10.17584556918669744480606496518, −9.523212258129501782557570627248, −7.64787651357487622629132176963, −7.18232029065375506572442901915, −6.06130131982933501843157234781, −4.88915496747179183013188222226, −3.62840635244523338886213703617, −1.56217856071047495262488104131, 1.12172030459422599498981706232, 2.94500929458780851993066463320, 3.87399692534106860523745302912, 5.28982848758469024487529566403, 6.62612544385312024326489112394, 8.019032367760528682351666713299, 8.849509050933797797623219628778, 10.20618571555384102830627736326, 11.18548074054614405562594566554, 12.11986759055479719736868997357

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