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Properties

Label 2-189-63.41-c3-0-13
Degree $2$
Conductor $189$
Sign $0.889 - 0.457i$
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  + (2.31 − 1.33i)2-s + (−0.417 + 0.723i)4-s + (−0.223 + 0.386i)5-s + (18.4 + 1.96i)7-s + 23.6i·8-s + 1.19i·10-s + (−34.2 + 19.7i)11-s + (68.4 + 39.5i)13-s + (45.3 − 20.0i)14-s + (28.3 + 49.0i)16-s − 9.74·17-s − 73.1i·19-s + (−0.186 − 0.323i)20-s + (−52.9 + 91.7i)22-s + (126. + 73.0i)23-s + ⋯
L(s)  = 1  + (0.819 − 0.473i)2-s + (−0.0522 + 0.0904i)4-s + (−0.0199 + 0.0345i)5-s + (0.994 + 0.106i)7-s + 1.04i·8-s + 0.0377i·10-s + (−0.939 + 0.542i)11-s + (1.46 + 0.843i)13-s + (0.865 − 0.383i)14-s + (0.442 + 0.766i)16-s − 0.139·17-s − 0.882i·19-s + (−0.00208 − 0.00361i)20-s + (−0.513 + 0.889i)22-s + (1.14 + 0.661i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 - 0.457i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.889 - 0.457i$
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: \((2,\ 189,\ (\ :3/2),\ 0.889 - 0.457i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.59893 + 0.628803i\)
\(L(\frac12)\) \(\approx\) \(2.59893 + 0.628803i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.4 - 1.96i)T \)
good2 \( 1 + (-2.31 + 1.33i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (0.223 - 0.386i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (34.2 - 19.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-68.4 - 39.5i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 9.74T + 4.91e3T^{2} \)
19 \( 1 + 73.1iT - 6.85e3T^{2} \)
23 \( 1 + (-126. - 73.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-134. + 77.4i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-9.87 - 5.70i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 337.T + 5.06e4T^{2} \)
41 \( 1 + (53.3 - 92.3i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (45.6 + 78.9i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (276. + 479. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 239. iT - 1.48e5T^{2} \)
59 \( 1 + (126. - 218. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (342. - 197. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-13.5 + 23.4i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 348. iT - 3.57e5T^{2} \)
73 \( 1 + 923. iT - 3.89e5T^{2} \)
79 \( 1 + (-280. - 485. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-281. - 487. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 644.T + 7.04e5T^{2} \)
97 \( 1 + (-427. + 246. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.15231409916994082346840867419, −11.31770599246509862032997343954, −10.71224187800166432757949521075, −9.005730396390981806735250298262, −8.242076844952793322464563493386, −6.98430543199736511779553841952, −5.33047123177782626548778550841, −4.59321306925687524777408415462, −3.25539914301255539237035353310, −1.79340590118785111655769803091, 1.03996844834528078227172848947, 3.26477376893487088790822363953, 4.65634130078318486003906494704, 5.51580588110466019825633672632, 6.52705022148399250293881812257, 7.957162648453665552584348350620, 8.730379362056367997552309828510, 10.43471941933265217170499744476, 10.86190082894990160480415506670, 12.35378777900953065640688150349

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