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Properties

Label 2-189-63.41-c3-0-18
Degree $2$
Conductor $189$
Sign $-0.885 + 0.465i$
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  + (1.10 − 0.638i)2-s + (−3.18 + 5.51i)4-s + (−1.59 + 2.76i)5-s + (−13.1 − 13.0i)7-s + 18.3i·8-s + 4.07i·10-s + (38.0 − 21.9i)11-s + (−65.6 − 37.8i)13-s + (−22.8 − 5.97i)14-s + (−13.7 − 23.8i)16-s − 104.·17-s − 74.7i·19-s + (−10.1 − 17.5i)20-s + (28.0 − 48.6i)22-s + (−46.6 − 26.9i)23-s + ⋯
L(s)  = 1  + (0.390 − 0.225i)2-s + (−0.398 + 0.689i)4-s + (−0.142 + 0.247i)5-s + (−0.711 − 0.702i)7-s + 0.810i·8-s + 0.128i·10-s + (1.04 − 0.602i)11-s + (−1.39 − 0.808i)13-s + (−0.436 − 0.114i)14-s + (−0.215 − 0.372i)16-s − 1.49·17-s − 0.902i·19-s + (−0.113 − 0.196i)20-s + (0.272 − 0.471i)22-s + (−0.422 − 0.244i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.465i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.885 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.885 + 0.465i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.885 + 0.465i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0890885 - 0.361137i\)
\(L(\frac12)\) \(\approx\) \(0.0890885 - 0.361137i\)
\(L(\frac{5}{2})\) 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 \( 1 + (13.1 + 13.0i)T \)
good2 \( 1 + (-1.10 + 0.638i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (1.59 - 2.76i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-38.0 + 21.9i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (65.6 + 37.8i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 104.T + 4.91e3T^{2} \)
19 \( 1 + 74.7iT - 6.85e3T^{2} \)
23 \( 1 + (46.6 + 26.9i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-26.3 + 15.2i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (111. + 64.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 46.1T + 5.06e4T^{2} \)
41 \( 1 + (213. - 369. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (166. + 287. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-170. - 294. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 235. iT - 1.48e5T^{2} \)
59 \( 1 + (-272. + 471. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (321. - 185. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (53.3 - 92.4i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 974. iT - 3.57e5T^{2} \)
73 \( 1 + 576. iT - 3.89e5T^{2} \)
79 \( 1 + (10.6 + 18.4i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-168. - 292. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.02e3T + 7.04e5T^{2} \)
97 \( 1 + (-63.5 + 36.6i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.75727547988862400705699099278, −10.93012573927234147180040935251, −9.647956538183602169444397871656, −8.734427168726524714638724100821, −7.46637957874042543340832840410, −6.57797590659869176888396140479, −4.89848444402645792968449447951, −3.81470620146395053349716077211, −2.72939106370620163131011833770, −0.13801629634405109698739712583, 2.00168032497335370139736676928, 4.02114110469585176078287253055, 4.96295148277613091849458486698, 6.27175836268373269793405626996, 7.01224751178545580867993092041, 8.845964751156393555139332012935, 9.435960365961243738191912825145, 10.34653738195046771928030510348, 11.92701764995062767949805324690, 12.43306806319698885074627016276

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