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Properties

Label 2-198-33.20-c2-0-4
Degree $2$
Conductor $198$
Sign $-0.0423 + 0.999i$
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  + (−1.34 − 0.437i)2-s + (1.61 + 1.17i)4-s + (−5.26 + 1.71i)5-s + (1.50 + 1.09i)7-s + (−1.66 − 2.28i)8-s + 7.83·10-s + (4.17 − 10.1i)11-s + (5.69 − 17.5i)13-s + (−1.54 − 2.12i)14-s + (1.23 + 3.80i)16-s + (−8.28 + 2.69i)17-s + (7.24 − 5.26i)19-s + (−10.5 − 3.42i)20-s + (−10.0 + 11.8i)22-s − 32.8i·23-s + ⋯
L(s)  = 1  + (−0.672 − 0.218i)2-s + (0.404 + 0.293i)4-s + (−1.05 + 0.342i)5-s + (0.215 + 0.156i)7-s + (−0.207 − 0.286i)8-s + 0.783·10-s + (0.379 − 0.925i)11-s + (0.437 − 1.34i)13-s + (−0.110 − 0.152i)14-s + (0.0772 + 0.237i)16-s + (−0.487 + 0.158i)17-s + (0.381 − 0.277i)19-s + (−0.526 − 0.171i)20-s + (−0.457 + 0.539i)22-s − 1.42i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0423 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0423 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $-0.0423 + 0.999i$
Analytic conductor: \(5.39510\)
Root analytic conductor: \(2.32273\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :1),\ -0.0423 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.513599 - 0.535845i\)
\(L(\frac12)\) \(\approx\) \(0.513599 - 0.535845i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 + 0.437i)T \)
3 \( 1 \)
11 \( 1 + (-4.17 + 10.1i)T \)
good5 \( 1 + (5.26 - 1.71i)T + (20.2 - 14.6i)T^{2} \)
7 \( 1 + (-1.50 - 1.09i)T + (15.1 + 46.6i)T^{2} \)
13 \( 1 + (-5.69 + 17.5i)T + (-136. - 99.3i)T^{2} \)
17 \( 1 + (8.28 - 2.69i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-7.24 + 5.26i)T + (111. - 343. i)T^{2} \)
23 \( 1 + 32.8iT - 529T^{2} \)
29 \( 1 + (-16.6 + 22.9i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-5.96 + 18.3i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (46.8 + 34.0i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-3.35 - 4.62i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 21.0T + 1.84e3T^{2} \)
47 \( 1 + (-35.0 - 48.2i)T + (-682. + 2.10e3i)T^{2} \)
53 \( 1 + (-77.9 - 25.3i)T + (2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (2.41 - 3.32i)T + (-1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (-15.5 - 47.9i)T + (-3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 + 17.7T + 4.48e3T^{2} \)
71 \( 1 + (35.1 - 11.4i)T + (4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (-24.5 - 17.8i)T + (1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (43.6 - 134. i)T + (-5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (45.4 - 14.7i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + 75.3iT - 7.92e3T^{2} \)
97 \( 1 + (-34.2 + 105. i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−11.72519518391858612234045604435, −11.02971392556122570176869935767, −10.22794694343441195660041984321, −8.717422379370542138161008713060, −8.184537689430409283253527696763, −7.09845147677082215773491193519, −5.82459796907337514707687940608, −4.03832738887571182198133645377, −2.83925010295777894952511655985, −0.56134494158833152312152541697, 1.57317665340748370638778821125, 3.77991489870722942997087178102, 4.94476908916536295403272520078, 6.69218185742408715883683417580, 7.43559261873960090623077047023, 8.539174312827126333503346675595, 9.331976626417655469495596504683, 10.49307340439030543529969829361, 11.76128845717478106932408322324, 11.95021701436283489734468447977

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