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Properties

Label 2-198-11.3-c1-0-2
Degree $2$
Conductor $198$
Sign $0.751 + 0.659i$
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 + (−1.11 − 0.812i)5-s + (0.881 − 2.71i)7-s + (0.309 + 0.951i)8-s + 1.38·10-s + (1.23 − 3.07i)11-s + (2 − 1.45i)13-s + (0.881 + 2.71i)14-s + (−0.809 − 0.587i)16-s + (−1 − 0.726i)17-s + (0.618 + 1.90i)19-s + (−1.11 + 0.812i)20-s + (0.809 + 3.21i)22-s + 7.23·23-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.154 − 0.475i)4-s + (−0.499 − 0.363i)5-s + (0.333 − 1.02i)7-s + (0.109 + 0.336i)8-s + 0.437·10-s + (0.372 − 0.927i)11-s + (0.554 − 0.403i)13-s + (0.235 + 0.725i)14-s + (−0.202 − 0.146i)16-s + (−0.242 − 0.176i)17-s + (0.141 + 0.436i)19-s + (−0.249 + 0.181i)20-s + (0.172 + 0.685i)22-s + 1.50·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $0.751 + 0.659i$
Analytic conductor: \(1.58103\)
Root analytic conductor: \(1.25739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :1/2),\ 0.751 + 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.797050 - 0.299938i\)
\(L(\frac12)\) \(\approx\) \(0.797050 - 0.299938i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 + 3.07i)T \)
good5 \( 1 + (1.11 + 0.812i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.881 + 2.71i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2 + 1.45i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1 + 0.726i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.618 - 1.90i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 7.23T + 23T^{2} \)
29 \( 1 + (-0.809 + 2.48i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.54 - 4.02i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3 - 9.23i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.47 + 10.6i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-2.85 - 8.78i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (9.39 - 6.82i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.02 - 3.16i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.85 + 4.25i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 6.94T + 67T^{2} \)
71 \( 1 + (-4.23 - 3.07i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.42 - 4.39i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-11.2 + 8.14i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-13.5 - 9.87i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 16.1T + 89T^{2} \)
97 \( 1 + (-11.2 + 8.14i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.27148494310578178532133279021, −11.08740162784455567174725652405, −10.56020256030906878460175849044, −9.162284379043155172523215513062, −8.312735434299910593703250692009, −7.43004214034121180913237154951, −6.31023827605380266914414257951, −4.91369638297219529135776454883, −3.54497369549426642834407718404, −0.995796626752168046257593757439, 1.98273167199173662093590558903, 3.52503629459513021413320407794, 5.03566855845480381745889196465, 6.63859898784036727474149954629, 7.61325758287043087564359745671, 8.845360532327267130823878930367, 9.421478710832780411680449788184, 10.87057905916140241769131185396, 11.46273111580763251633745887174, 12.35037973862495305251992296748

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