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Properties

Label 2-189-63.59-c3-0-1
Degree $2$
Conductor $189$
Sign $-0.883 - 0.467i$
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  + (0.998 + 0.576i)2-s + (−3.33 − 5.77i)4-s − 0.274·5-s + (−9.15 + 16.1i)7-s − 16.9i·8-s + (−0.274 − 0.158i)10-s + 26.2i·11-s + (−39.6 − 22.8i)13-s + (−18.4 + 10.7i)14-s + (−16.9 + 29.3i)16-s + (−30.2 + 52.3i)17-s + (−38.8 + 22.4i)19-s + (0.916 + 1.58i)20-s + (−15.1 + 26.1i)22-s + 127. i·23-s + ⋯
L(s)  = 1  + (0.352 + 0.203i)2-s + (−0.416 − 0.722i)4-s − 0.0245·5-s + (−0.494 + 0.869i)7-s − 0.747i·8-s + (−0.00867 − 0.00500i)10-s + 0.719i·11-s + (−0.845 − 0.487i)13-s + (−0.351 + 0.206i)14-s + (−0.264 + 0.458i)16-s + (−0.431 + 0.747i)17-s + (−0.469 + 0.271i)19-s + (0.0102 + 0.0177i)20-s + (−0.146 + 0.253i)22-s + 1.15i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.883 - 0.467i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.883 - 0.467i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.103629 + 0.417521i\)
\(L(\frac12)\) \(\approx\) \(0.103629 + 0.417521i\)
\(L(\frac{5}{2})\) 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 \( 1 + (9.15 - 16.1i)T \)
good2 \( 1 + (-0.998 - 0.576i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + 0.274T + 125T^{2} \)
11 \( 1 - 26.2iT - 1.33e3T^{2} \)
13 \( 1 + (39.6 + 22.8i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (30.2 - 52.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (38.8 - 22.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 - 127. iT - 1.21e4T^{2} \)
29 \( 1 + (58.2 - 33.6i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-85.6 + 49.4i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (123. + 214. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (134. - 232. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (72.4 + 125. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-250. + 434. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-333. - 192. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (274. + 475. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-189. - 109. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-378. - 656. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.10e3iT - 3.57e5T^{2} \)
73 \( 1 + (125. + 72.6i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-445. + 772. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (745. + 1.29e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-145. - 251. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-144. + 83.6i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.69521717583962631644735322490, −11.73399314917486641229129781792, −10.28168604198096757272143557319, −9.674950614135487573318327346732, −8.641159166567946026112552984818, −7.21432301047151113386577831450, −6.01111500527365608690770549992, −5.20452291843943004230814441732, −3.86926871507643068307735582966, −2.03859231248080589910039697895, 0.16076250518408842453138341203, 2.64361229942183918814360634650, 3.91255114056196218777864350946, 4.87302767692015795500777462976, 6.53222530809837081051366422582, 7.56941513381257599810431840909, 8.661768739761642917372242164776, 9.684498330379634106996647422094, 10.85405396231982491137780726937, 11.84838263149836865874231832154

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