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Properties

Label 2-198-11.5-c3-0-5
Degree $2$
Conductor $198$
Sign $-0.544 - 0.838i$
Analytic cond. $11.6823$
Root an. cond. $3.41794$
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  + (0.618 + 1.90i)2-s + (−3.23 + 2.35i)4-s + (−1.67 + 5.14i)5-s + (17.9 − 13.0i)7-s + (−6.47 − 4.70i)8-s − 10.8·10-s + (11.0 + 34.7i)11-s + (23.7 + 73.0i)13-s + (35.8 + 26.0i)14-s + (4.94 − 15.2i)16-s + (−18.3 + 56.3i)17-s + (−77.0 − 55.9i)19-s + (−6.68 − 20.5i)20-s + (−59.3 + 42.5i)22-s − 142.·23-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.404 + 0.293i)4-s + (−0.149 + 0.460i)5-s + (0.967 − 0.702i)7-s + (−0.286 − 0.207i)8-s − 0.342·10-s + (0.302 + 0.953i)11-s + (0.506 + 1.55i)13-s + (0.684 + 0.497i)14-s + (0.0772 − 0.237i)16-s + (−0.261 + 0.804i)17-s + (−0.930 − 0.676i)19-s + (−0.0747 − 0.230i)20-s + (−0.574 + 0.411i)22-s − 1.29·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.544 - 0.838i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.544 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $-0.544 - 0.838i$
Analytic conductor: \(11.6823\)
Root analytic conductor: \(3.41794\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :3/2),\ -0.544 - 0.838i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.845145 + 1.55687i\)
\(L(\frac12)\) \(\approx\) \(0.845145 + 1.55687i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.618 - 1.90i)T \)
3 \( 1 \)
11 \( 1 + (-11.0 - 34.7i)T \)
good5 \( 1 + (1.67 - 5.14i)T + (-101. - 73.4i)T^{2} \)
7 \( 1 + (-17.9 + 13.0i)T + (105. - 326. i)T^{2} \)
13 \( 1 + (-23.7 - 73.0i)T + (-1.77e3 + 1.29e3i)T^{2} \)
17 \( 1 + (18.3 - 56.3i)T + (-3.97e3 - 2.88e3i)T^{2} \)
19 \( 1 + (77.0 + 55.9i)T + (2.11e3 + 6.52e3i)T^{2} \)
23 \( 1 + 142.T + 1.21e4T^{2} \)
29 \( 1 + (16.5 - 12.0i)T + (7.53e3 - 2.31e4i)T^{2} \)
31 \( 1 + (-65.9 - 202. i)T + (-2.41e4 + 1.75e4i)T^{2} \)
37 \( 1 + (-117. + 85.5i)T + (1.56e4 - 4.81e4i)T^{2} \)
41 \( 1 + (-67.0 - 48.6i)T + (2.12e4 + 6.55e4i)T^{2} \)
43 \( 1 + 151.T + 7.95e4T^{2} \)
47 \( 1 + (-73.1 - 53.1i)T + (3.20e4 + 9.87e4i)T^{2} \)
53 \( 1 + (-72.5 - 223. i)T + (-1.20e5 + 8.75e4i)T^{2} \)
59 \( 1 + (-244. + 177. i)T + (6.34e4 - 1.95e5i)T^{2} \)
61 \( 1 + (46.1 - 141. i)T + (-1.83e5 - 1.33e5i)T^{2} \)
67 \( 1 - 826.T + 3.00e5T^{2} \)
71 \( 1 + (-277. + 854. i)T + (-2.89e5 - 2.10e5i)T^{2} \)
73 \( 1 + (-111. + 81.1i)T + (1.20e5 - 3.69e5i)T^{2} \)
79 \( 1 + (94.0 + 289. i)T + (-3.98e5 + 2.89e5i)T^{2} \)
83 \( 1 + (-236. + 726. i)T + (-4.62e5 - 3.36e5i)T^{2} \)
89 \( 1 - 313.T + 7.04e5T^{2} \)
97 \( 1 + (180. + 553. i)T + (-7.38e5 + 5.36e5i)T^{2} \)
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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

−12.40808330198454206501314984427, −11.35592102897975872847677763668, −10.51541579612038321520408794870, −9.183646918766094859810153846572, −8.177576379244600734385061944149, −7.08583825536591789428971941841, −6.40450423433090794946875371215, −4.67485596278319877328169084163, −4.00078545056211054302871338701, −1.80581440364272826723430779815, 0.75316968661260623291676758183, 2.40218465819802130724660017286, 3.90073815411787567737681251929, 5.19468886477606543335421489984, 6.07333426783036323049305013995, 8.182534862789719428782642717871, 8.455291377917665292838131813569, 9.857731023634986112461072513230, 10.94532153503084011944012511359, 11.66904032012336952655929786087

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