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Properties

Label 2-198-11.8-c2-0-7
Degree $2$
Conductor $198$
Sign $-0.318 + 0.947i$
Analytic cond. $5.39510$
Root an. cond. $2.32273$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.831 + 1.14i)2-s + (−0.618 + 1.90i)4-s + (−6.27 − 4.56i)5-s + (−2.67 − 0.869i)7-s + (−2.68 + 0.874i)8-s − 10.9i·10-s + (−2.55 − 10.6i)11-s + (−5.81 − 8.00i)13-s + (−1.23 − 3.78i)14-s + (−3.23 − 2.35i)16-s + (4.12 − 5.67i)17-s + (−16.8 + 5.46i)19-s + (12.5 − 9.12i)20-s + (10.1 − 11.8i)22-s − 17.7·23-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (−0.154 + 0.475i)4-s + (−1.25 − 0.912i)5-s + (−0.382 − 0.124i)7-s + (−0.336 + 0.109i)8-s − 1.09i·10-s + (−0.232 − 0.972i)11-s + (−0.447 − 0.615i)13-s + (−0.0878 − 0.270i)14-s + (−0.202 − 0.146i)16-s + (0.242 − 0.333i)17-s + (−0.885 + 0.287i)19-s + (0.627 − 0.456i)20-s + (0.459 − 0.537i)22-s − 0.772·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.318 + 0.947i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $-0.318 + 0.947i$
Analytic conductor: \(5.39510\)
Root analytic conductor: \(2.32273\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :1),\ -0.318 + 0.947i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.366874 - 0.510502i\)
\(L(\frac12)\) \(\approx\) \(0.366874 - 0.510502i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.831 - 1.14i)T \)
3 \( 1 \)
11 \( 1 + (2.55 + 10.6i)T \)
good5 \( 1 + (6.27 + 4.56i)T + (7.72 + 23.7i)T^{2} \)
7 \( 1 + (2.67 + 0.869i)T + (39.6 + 28.8i)T^{2} \)
13 \( 1 + (5.81 + 8.00i)T + (-52.2 + 160. i)T^{2} \)
17 \( 1 + (-4.12 + 5.67i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (16.8 - 5.46i)T + (292. - 212. i)T^{2} \)
23 \( 1 + 17.7T + 529T^{2} \)
29 \( 1 + (-29.6 - 9.62i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (-9.04 + 6.56i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (-6.14 + 18.9i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (76.0 - 24.7i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 - 20.7iT - 1.84e3T^{2} \)
47 \( 1 + (-8.30 - 25.5i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-52.0 + 37.8i)T + (868. - 2.67e3i)T^{2} \)
59 \( 1 + (-11.0 + 34.1i)T + (-2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (-30.4 + 41.9i)T + (-1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 + 17.1T + 4.48e3T^{2} \)
71 \( 1 + (-25.5 - 18.5i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-2.48 - 0.805i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (-47.3 - 65.2i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (31.8 - 43.8i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 + 85.3T + 7.92e3T^{2} \)
97 \( 1 + (-12.3 + 8.96i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−12.19809584328100459874170536698, −11.25250025728094824925822817736, −9.906857796089877162358100813565, −8.390272655424424568588990827963, −8.132172386158185547044409693881, −6.78898934084965325176892096367, −5.47643389236039385278166836679, −4.38031902227985996843297287327, −3.26254205093931740021131321855, −0.30440852266994923356007442354, 2.39631481174491603943336745430, 3.70846848496909281477896610591, 4.68175411743877936945601050057, 6.44120353184729290781907907730, 7.30979772544018959301699294562, 8.526753252284413056569146289168, 9.987601968477039756798217818157, 10.61700676975894846654612708953, 11.89379912846313380704734599575, 12.12693893764380383695438544127

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