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Properties

Label 2-189-9.4-c3-0-16
Degree $2$
Conductor $189$
Sign $0.140 + 0.990i$
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  + (0.403 + 0.699i)2-s + (3.67 − 6.36i)4-s + (9.11 − 15.7i)5-s + (3.5 + 6.06i)7-s + 12.3·8-s + 14.7·10-s + (−24.7 − 42.8i)11-s + (−22.2 + 38.5i)13-s + (−2.82 + 4.89i)14-s + (−24.3 − 42.2i)16-s − 47.2·17-s + 56.7·19-s + (−66.9 − 116. i)20-s + (19.9 − 34.5i)22-s + (−27.7 + 47.9i)23-s + ⋯
L(s)  = 1  + (0.142 + 0.247i)2-s + (0.459 − 0.795i)4-s + (0.815 − 1.41i)5-s + (0.188 + 0.327i)7-s + 0.547·8-s + 0.465·10-s + (−0.677 − 1.17i)11-s + (−0.474 + 0.821i)13-s + (−0.0539 + 0.0934i)14-s + (−0.381 − 0.660i)16-s − 0.673·17-s + 0.685·19-s + (−0.748 − 1.29i)20-s + (0.193 − 0.334i)22-s + (−0.251 + 0.435i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.140 + 0.990i$
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: \((2,\ 189,\ (\ :3/2),\ 0.140 + 0.990i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.68066 - 1.45843i\)
\(L(\frac12)\) \(\approx\) \(1.68066 - 1.45843i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.403 - 0.699i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-9.11 + 15.7i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (24.7 + 42.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (22.2 - 38.5i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 47.2T + 4.91e3T^{2} \)
19 \( 1 - 56.7T + 6.85e3T^{2} \)
23 \( 1 + (27.7 - 47.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-75.1 - 130. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-83.7 + 145. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 331.T + 5.06e4T^{2} \)
41 \( 1 + (-147. + 255. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-158. - 275. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (68.9 + 119. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 411.T + 1.48e5T^{2} \)
59 \( 1 + (-106. + 183. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-26.1 - 45.3i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (353. - 612. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 78.7T + 3.57e5T^{2} \)
73 \( 1 - 839.T + 3.89e5T^{2} \)
79 \( 1 + (507. + 879. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-543. - 941. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 762.T + 7.04e5T^{2} \)
97 \( 1 + (-671. - 1.16e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.85797253939022962860465930695, −10.95130314315974701738126631180, −9.703311228556170078949564369999, −9.035732947517284611775282270387, −7.85702336284423820950757788368, −6.29830532171398271299077320838, −5.49162188584685105525946453498, −4.66679878604099126334505568234, −2.29885314066995785332676905321, −0.950499627761113563824491703974, 2.22913061501501026012279088946, 2.99262691425865275522978630260, 4.61776730868757114484924814590, 6.25725774718927943135102302545, 7.22922019635119156321231836781, 7.918870146458454126926381212603, 9.742475311676006983953125098692, 10.45138317008380038181819967602, 11.22056876740242839518793467732, 12.37515359734844701861667901385

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