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Properties

Label 2-189-63.5-c3-0-4
Degree $2$
Conductor $189$
Sign $-0.255 - 0.966i$
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  + 3.72i·2-s − 5.84·4-s + (−6.83 − 11.8i)5-s + (18.4 − 0.984i)7-s + 8.02i·8-s + (44.0 − 25.4i)10-s + (21.3 + 12.3i)11-s + (2.03 + 1.17i)13-s + (3.66 + 68.8i)14-s − 76.6·16-s + (63.8 + 110. i)17-s + (87.2 + 50.3i)19-s + (39.9 + 69.1i)20-s + (−45.7 + 79.2i)22-s + (−42.0 + 24.2i)23-s + ⋯
L(s)  = 1  + 1.31i·2-s − 0.730·4-s + (−0.611 − 1.05i)5-s + (0.998 − 0.0531i)7-s + 0.354i·8-s + (1.39 − 0.803i)10-s + (0.583 + 0.337i)11-s + (0.0434 + 0.0250i)13-s + (0.0699 + 1.31i)14-s − 1.19·16-s + (0.910 + 1.57i)17-s + (1.05 + 0.608i)19-s + (0.446 + 0.773i)20-s + (−0.443 + 0.768i)22-s + (−0.381 + 0.219i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.255 - 0.966i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.255 - 0.966i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.255 - 0.966i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.11195 + 1.44346i\)
\(L(\frac12)\) \(\approx\) \(1.11195 + 1.44346i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.4 + 0.984i)T \)
good2 \( 1 - 3.72iT - 8T^{2} \)
5 \( 1 + (6.83 + 11.8i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-21.3 - 12.3i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-2.03 - 1.17i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-63.8 - 110. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-87.2 - 50.3i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (42.0 - 24.2i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-171. + 99.0i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 42.9iT - 2.97e4T^{2} \)
37 \( 1 + (42.0 - 72.7i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (92.8 - 160. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (185. + 321. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 504.T + 1.03e5T^{2} \)
53 \( 1 + (372. - 215. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 312.T + 2.05e5T^{2} \)
61 \( 1 - 548. iT - 2.26e5T^{2} \)
67 \( 1 + 651.T + 3.00e5T^{2} \)
71 \( 1 + 18.9iT - 3.57e5T^{2} \)
73 \( 1 + (-525. + 303. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 435.T + 4.93e5T^{2} \)
83 \( 1 + (424. + 734. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-147. + 254. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-900. + 520. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.23061787054060704942853549712, −11.77147822843020183630420902050, −10.30050735703447679139517585903, −8.826056090796359560372713699738, −8.152686633842069460719141196858, −7.49189903521856060073028723971, −6.06196776987331865962759623514, −5.07237678659646644369359893884, −4.06957775903300919473636034605, −1.40500355886046447213875355838, 0.975712698532693469065243013940, 2.66330338500950795973627250271, 3.58229337496317975762259348688, 4.98352184936183562154645173658, 6.80310343458673411900722845592, 7.70635886800859489218892086713, 9.129393071604657337736194007037, 10.17595877092631385904468990559, 11.15304173224422804822103332332, 11.58846873994505483506084353331

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