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Properties

Label 2-198-11.4-c1-0-4
Degree $2$
Conductor $198$
Sign $0.970 - 0.242i$
Analytic cond. $1.58103$
Root an. cond. $1.25739$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (2.61 − 1.90i)5-s + (−0.618 − 1.90i)7-s + (−0.309 + 0.951i)8-s + 3.23·10-s + (−0.309 + 3.30i)11-s + (−1 − 0.726i)13-s + (0.618 − 1.90i)14-s + (−0.809 + 0.587i)16-s + (−0.5 + 0.363i)17-s + (−1.80 + 5.56i)19-s + (2.61 + 1.90i)20-s + (−2.19 + 2.48i)22-s − 1.23·23-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.154 + 0.475i)4-s + (1.17 − 0.850i)5-s + (−0.233 − 0.718i)7-s + (−0.109 + 0.336i)8-s + 1.02·10-s + (−0.0931 + 0.995i)11-s + (−0.277 − 0.201i)13-s + (0.165 − 0.508i)14-s + (−0.202 + 0.146i)16-s + (−0.121 + 0.0881i)17-s + (−0.415 + 1.27i)19-s + (0.585 + 0.425i)20-s + (−0.467 + 0.530i)22-s − 0.257·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(1.58103\)
Root analytic conductor: \(1.25739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :1/2),\ 0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74710 + 0.214765i\)
\(L(\frac12)\) \(\approx\) \(1.74710 + 0.214765i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 \)
11 \( 1 + (0.309 - 3.30i)T \)
good5 \( 1 + (-2.61 + 1.90i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.618 + 1.90i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (1 + 0.726i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.5 - 0.363i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.80 - 5.56i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 1.23T + 23T^{2} \)
29 \( 1 + (1.38 + 4.25i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.61 + 1.17i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.14 + 3.52i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.73 - 5.34i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 8.56T + 43T^{2} \)
47 \( 1 + (-2 + 6.15i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.23 + 0.898i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.66 - 8.19i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2 + 1.45i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 11.0T + 67T^{2} \)
71 \( 1 + (4.23 - 3.07i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.20 - 9.87i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (10.8 + 7.88i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-7.54 + 5.48i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 8.09T + 89T^{2} \)
97 \( 1 + (5.78 + 4.20i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.85778931440903124267193145800, −11.89491960873848030105834595051, −10.28319942781557226512839404412, −9.709180311757863304071741008629, −8.421151471321981263040546367623, −7.26087293608872918880696800607, −6.10708064878690154466764155187, −5.12886967557345448500791312390, −3.98986234558101760152962568396, −1.97309528593322042496544463947, 2.21559947784590930761765738153, 3.22099036196459387699872880044, 5.10268009057880848750849638668, 6.07580504298411429383273445607, 6.88883756305908217666866586092, 8.728251995297746091194961808069, 9.638848451370248398739133571960, 10.65178934767464467858588114019, 11.37437676965203330283132372062, 12.55524087777361345543805187766

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