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Properties

Label 2-198-11.8-c2-0-4
Degree $2$
Conductor $198$
Sign $0.0225 - 0.999i$
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  + (0.831 + 1.14i)2-s + (−0.618 + 1.90i)4-s + (7.34 + 5.33i)5-s + (−1.23 − 0.401i)7-s + (−2.68 + 0.874i)8-s + 12.8i·10-s + (−5.62 − 9.45i)11-s + (11.5 + 15.9i)13-s + (−0.567 − 1.74i)14-s + (−3.23 − 2.35i)16-s + (6.34 − 8.73i)17-s + (−31.5 + 10.2i)19-s + (−14.6 + 10.6i)20-s + (6.14 − 14.2i)22-s + 27.4·23-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (−0.154 + 0.475i)4-s + (1.46 + 1.06i)5-s + (−0.176 − 0.0573i)7-s + (−0.336 + 0.109i)8-s + 1.28i·10-s + (−0.510 − 0.859i)11-s + (0.890 + 1.22i)13-s + (−0.0405 − 0.124i)14-s + (−0.202 − 0.146i)16-s + (0.373 − 0.513i)17-s + (−1.65 + 0.538i)19-s + (−0.734 + 0.533i)20-s + (0.279 − 0.649i)22-s + 1.19·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0225 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0225 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $0.0225 - 0.999i$
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: \((2,\ 198,\ (\ :1),\ 0.0225 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.57208 + 1.53706i\)
\(L(\frac12)\) \(\approx\) \(1.57208 + 1.53706i\)
\(L(2)\) 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 \( 1 + (5.62 + 9.45i)T \)
good5 \( 1 + (-7.34 - 5.33i)T + (7.72 + 23.7i)T^{2} \)
7 \( 1 + (1.23 + 0.401i)T + (39.6 + 28.8i)T^{2} \)
13 \( 1 + (-11.5 - 15.9i)T + (-52.2 + 160. i)T^{2} \)
17 \( 1 + (-6.34 + 8.73i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (31.5 - 10.2i)T + (292. - 212. i)T^{2} \)
23 \( 1 - 27.4T + 529T^{2} \)
29 \( 1 + (7.99 + 2.59i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (-3.61 + 2.62i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (-15.3 + 47.2i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (-2.99 + 0.974i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 + 2.29iT - 1.84e3T^{2} \)
47 \( 1 + (18.9 + 58.3i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-52.3 + 38.0i)T + (868. - 2.67e3i)T^{2} \)
59 \( 1 + (-12.8 + 39.5i)T + (-2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (-24.3 + 33.5i)T + (-1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 - 22.6T + 4.48e3T^{2} \)
71 \( 1 + (-71.1 - 51.7i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (84.0 + 27.3i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (-22.8 - 31.5i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-30.4 + 41.8i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 - 47.5T + 7.92e3T^{2} \)
97 \( 1 + (110. - 80.5i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−12.97769864277243776231834285728, −11.33588993278834820193331190745, −10.60638510370728167500456802606, −9.482857109257043900861242474474, −8.496719920622893953176590642742, −6.92226590363742807230613870645, −6.30978459059088410180176723759, −5.38991603128830724588004669563, −3.62754873161200969303981198924, −2.23933604036690168725641699263, 1.28465852337024008472909525851, 2.67219838921520918469845809372, 4.54349675380537881664821185746, 5.48825410165204469511359189569, 6.39549347995644235116502965537, 8.280840184989766838453291217818, 9.205562008032582057161989114359, 10.15948491841005875609319053891, 10.82851721480104196870912117781, 12.45753585338675574265999360299

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