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Properties

Label 2-198-11.9-c1-0-2
Degree $2$
Conductor $198$
Sign $0.998 + 0.0475i$
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  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (−1.11 − 3.44i)5-s + (3.11 + 2.26i)7-s + (0.809 − 0.587i)8-s + 3.61·10-s + (3.23 − 0.726i)11-s + (2 − 6.15i)13-s + (−3.11 + 2.26i)14-s + (0.309 + 0.951i)16-s + (1 + 3.07i)17-s + (−1.61 + 1.17i)19-s + (−1.11 + 3.44i)20-s + (−0.309 + 3.30i)22-s − 2.76·23-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.404 − 0.293i)4-s + (−0.499 − 1.53i)5-s + (1.17 + 0.856i)7-s + (0.286 − 0.207i)8-s + 1.14·10-s + (0.975 − 0.219i)11-s + (0.554 − 1.70i)13-s + (−0.833 + 0.605i)14-s + (0.0772 + 0.237i)16-s + (0.242 + 0.746i)17-s + (−0.371 + 0.269i)19-s + (−0.249 + 0.769i)20-s + (−0.0658 + 0.704i)22-s − 0.576·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $0.998 + 0.0475i$
Analytic conductor: \(1.58103\)
Root analytic conductor: \(1.25739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :1/2),\ 0.998 + 0.0475i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08925 - 0.0259008i\)
\(L(\frac12)\) \(\approx\) \(1.08925 - 0.0259008i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 \)
11 \( 1 + (-3.23 + 0.726i)T \)
good5 \( 1 + (1.11 + 3.44i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-3.11 - 2.26i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-2 + 6.15i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1 - 3.07i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.61 - 1.17i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 2.76T + 23T^{2} \)
29 \( 1 + (-0.309 - 0.224i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.0450 + 0.138i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3 + 2.17i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (5.47 - 3.97i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-3.85 + 2.80i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.89 - 8.92i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-9.97 - 7.24i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.854 - 2.62i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 + (-0.236 - 0.726i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.92 - 1.40i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.20 - 6.79i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.40 + 7.41i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 6.18T + 89T^{2} \)
97 \( 1 + (2.20 - 6.79i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.42757910320083668857844368751, −11.73980092392983942308439195381, −10.43928618868272702320049080610, −8.963042551721712730657385321814, −8.426306246754472259870341099665, −7.83924838666534682210502901557, −5.93485857743465800108987064165, −5.20529226646815673217827797006, −4.02120569378425752305914967712, −1.29300898395957009285834915342, 1.86771694204349570500830798011, 3.61951541858502435670605095092, 4.47547007380972745810882099586, 6.64720555501073749859461749189, 7.31037755786952683107297341345, 8.521116798764708632627527910969, 9.767216732228450968755224081442, 10.81024894927618840005403405194, 11.42690647583777444648603509417, 11.91634026248630549200986986147

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