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Properties

Label 2-189-63.38-c3-0-15
Degree $2$
Conductor $189$
Sign $0.880 - 0.473i$
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  + 4.92i·2-s − 16.2·4-s + (4.62 − 8.00i)5-s + (−13.0 − 13.1i)7-s − 40.8i·8-s + (39.4 + 22.7i)10-s + (−13.2 + 7.62i)11-s + (61.0 − 35.2i)13-s + (64.6 − 64.4i)14-s + 71.0·16-s + (17.8 − 30.9i)17-s + (91.6 − 52.9i)19-s + (−75.3 + 130. i)20-s + (−37.6 − 65.1i)22-s + (80.0 + 46.2i)23-s + ⋯
L(s)  = 1  + 1.74i·2-s − 2.03·4-s + (0.413 − 0.716i)5-s + (−0.705 − 0.708i)7-s − 1.80i·8-s + (1.24 + 0.720i)10-s + (−0.362 + 0.209i)11-s + (1.30 − 0.752i)13-s + (1.23 − 1.22i)14-s + 1.11·16-s + (0.255 − 0.441i)17-s + (1.10 − 0.638i)19-s + (−0.842 + 1.45i)20-s + (−0.364 − 0.631i)22-s + (0.725 + 0.419i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.880 - 0.473i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.880 - 0.473i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.29606 + 0.326304i\)
\(L(\frac12)\) \(\approx\) \(1.29606 + 0.326304i\)
\(L(\frac{5}{2})\) 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 \( 1 + (13.0 + 13.1i)T \)
good2 \( 1 - 4.92iT - 8T^{2} \)
5 \( 1 + (-4.62 + 8.00i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (13.2 - 7.62i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-61.0 + 35.2i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-17.8 + 30.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-91.6 + 52.9i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-80.0 - 46.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (222. + 128. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 142. iT - 2.97e4T^{2} \)
37 \( 1 + (100. + 173. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (162. + 280. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (92.5 - 160. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 471.T + 1.03e5T^{2} \)
53 \( 1 + (247. + 142. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 170.T + 2.05e5T^{2} \)
61 \( 1 + 257. iT - 2.26e5T^{2} \)
67 \( 1 - 1.00e3T + 3.00e5T^{2} \)
71 \( 1 + 445. iT - 3.57e5T^{2} \)
73 \( 1 + (81.3 + 46.9i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 463.T + 4.93e5T^{2} \)
83 \( 1 + (348. - 603. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (338. + 586. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (402. + 232. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.86084758358049930454151271026, −11.07166565007086699167481374881, −9.654507035439006930756041335374, −9.042592908885660441386937805963, −7.83080126674990063459091081416, −7.07447191135184521931700258814, −5.84397227484060890517648646515, −5.15120635420783483346973722589, −3.66849308544142818971579107792, −0.65206537636519207441566550693, 1.48446557663588290054070194801, 2.87333117486241139600483675922, 3.66616767484404503289424365825, 5.41845579929193043041189088487, 6.66105954698478679847270570198, 8.547123284613879797620330722733, 9.371532359118085347629853069974, 10.30306120609143980729514721912, 11.01180505426529497659902717896, 11.92039295320379405805418630181

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