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Properties

Label 2-189-63.5-c3-0-0
Degree $2$
Conductor $189$
Sign $0.419 + 0.907i$
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  + 5.38i·2-s − 20.9·4-s + (5.57 + 9.66i)5-s + (−8.46 + 16.4i)7-s − 69.7i·8-s + (−52.0 + 30.0i)10-s + (−18.1 − 10.4i)11-s + (−46.0 − 26.5i)13-s + (−88.6 − 45.5i)14-s + 207.·16-s + (44.2 + 76.6i)17-s + (−40.4 − 23.3i)19-s + (−116. − 202. i)20-s + (56.2 − 97.4i)22-s + (68.0 − 39.3i)23-s + ⋯
L(s)  = 1  + 1.90i·2-s − 2.61·4-s + (0.499 + 0.864i)5-s + (−0.457 + 0.889i)7-s − 3.08i·8-s + (−1.64 + 0.949i)10-s + (−0.496 − 0.286i)11-s + (−0.981 − 0.566i)13-s + (−1.69 − 0.869i)14-s + 3.24·16-s + (0.631 + 1.09i)17-s + (−0.488 − 0.281i)19-s + (−1.30 − 2.26i)20-s + (0.545 − 0.944i)22-s + (0.617 − 0.356i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.419 + 0.907i$
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: \((2,\ 189,\ (\ :3/2),\ 0.419 + 0.907i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.526413 - 0.336764i\)
\(L(\frac12)\) \(\approx\) \(0.526413 - 0.336764i\)
\(L(\frac{5}{2})\) 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 \( 1 + (8.46 - 16.4i)T \)
good2 \( 1 - 5.38iT - 8T^{2} \)
5 \( 1 + (-5.57 - 9.66i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (18.1 + 10.4i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (46.0 + 26.5i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-44.2 - 76.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (40.4 + 23.3i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-68.0 + 39.3i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-17.2 + 9.97i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 28.0iT - 2.97e4T^{2} \)
37 \( 1 + (122. - 211. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (16.9 - 29.3i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-87.5 - 151. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 146.T + 1.03e5T^{2} \)
53 \( 1 + (656. - 378. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 83.9T + 2.05e5T^{2} \)
61 \( 1 + 92.1iT - 2.26e5T^{2} \)
67 \( 1 + 897.T + 3.00e5T^{2} \)
71 \( 1 + 706. iT - 3.57e5T^{2} \)
73 \( 1 + (529. - 305. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + 266.T + 4.93e5T^{2} \)
83 \( 1 + (-311. - 539. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (355. - 615. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (498. - 287. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.16631859728325649258673224102, −12.44239620660459778735987896537, −10.50401391168042064607675156490, −9.663913239812329405866250630727, −8.586334641589451482670176633282, −7.68361878587691636332951407717, −6.53511090911216206266370001633, −5.91817922743630025656918138148, −4.86829538936372393819203936109, −3.00497793710117694183808895966, 0.27871972516137515001819013207, 1.64719529226126339664482031152, 3.05904359855241995509141597427, 4.43758844896198632683869674035, 5.25541860458768007000837407768, 7.39558442019032762157396374343, 8.887014098001496937078693628797, 9.657710402432446796192587900921, 10.25467589797928947337454722375, 11.33147086489494640040642428438

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