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Properties

Label 2-189-63.58-c3-0-9
Degree $2$
Conductor $189$
Sign $0.154 - 0.988i$
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  + 2.37·2-s − 2.34·4-s + (9.23 + 15.9i)5-s + (15.8 − 9.54i)7-s − 24.5·8-s + (21.9 + 38.0i)10-s + (−28.3 + 49.1i)11-s + (3.61 − 6.26i)13-s + (37.7 − 22.6i)14-s − 39.7·16-s + (42.2 + 73.2i)17-s + (−1.77 + 3.07i)19-s + (−21.6 − 37.4i)20-s + (−67.5 + 116. i)22-s + (45.3 + 78.5i)23-s + ⋯
L(s)  = 1  + 0.841·2-s − 0.292·4-s + (0.825 + 1.43i)5-s + (0.857 − 0.515i)7-s − 1.08·8-s + (0.694 + 1.20i)10-s + (−0.777 + 1.34i)11-s + (0.0772 − 0.133i)13-s + (0.720 − 0.433i)14-s − 0.621·16-s + (0.603 + 1.04i)17-s + (−0.0214 + 0.0371i)19-s + (−0.241 − 0.418i)20-s + (−0.654 + 1.13i)22-s + (0.411 + 0.711i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.154 - 0.988i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.154 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.154 - 0.988i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.154 - 0.988i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.88910 + 1.61745i\)
\(L(\frac12)\) \(\approx\) \(1.88910 + 1.61745i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-15.8 + 9.54i)T \)
good2 \( 1 - 2.37T + 8T^{2} \)
5 \( 1 + (-9.23 - 15.9i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (28.3 - 49.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-3.61 + 6.26i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-42.2 - 73.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (1.77 - 3.07i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-45.3 - 78.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (25.6 + 44.4i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 156.T + 2.97e4T^{2} \)
37 \( 1 + (-28.9 + 50.1i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-79.8 + 138. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (20.6 + 35.7i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 144.T + 1.03e5T^{2} \)
53 \( 1 + (369. + 639. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 225.T + 2.05e5T^{2} \)
61 \( 1 + 267.T + 2.26e5T^{2} \)
67 \( 1 - 894.T + 3.00e5T^{2} \)
71 \( 1 - 52.1T + 3.57e5T^{2} \)
73 \( 1 + (289. + 501. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 495.T + 4.93e5T^{2} \)
83 \( 1 + (380. + 658. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (364. - 631. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (565. + 979. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.59961935825901427250183981403, −11.34857021530043613348083023892, −10.34335436802486001076249348322, −9.720803114800418425665723486987, −8.057865215250420517642722492470, −6.99449366818562245404435114800, −5.83679703501671167910198208366, −4.79855845989842531418508810449, −3.47451122196542470526500118612, −2.08368481097332883134635446769, 0.890389913746334420947128827698, 2.80457390483784191683158154483, 4.62996855450449555048523970414, 5.26174305582369656882323953352, 6.00991838335821587822318507210, 8.164958043579784376133927302131, 8.789139584637906671507892790571, 9.680746430154772357509007922575, 11.20492790251249804460181397347, 12.21076065889221518768942208914

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