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Properties

Label 2-198-11.8-c2-0-1
Degree $2$
Conductor $198$
Sign $0.981 + 0.189i$
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 + (2.42 + 1.76i)5-s + (−3.70 − 1.20i)7-s + (2.68 − 0.874i)8-s − 4.23i·10-s + (10.2 − 3.96i)11-s + (14.4 + 19.8i)13-s + (1.70 + 5.23i)14-s + (−3.23 − 2.35i)16-s + (−3.79 + 5.22i)17-s + (14.9 − 4.84i)19-s + (−4.85 + 3.52i)20-s + (−13.0 − 8.44i)22-s + 30.3·23-s + ⋯
L(s)  = 1  + (−0.415 − 0.572i)2-s + (−0.154 + 0.475i)4-s + (0.485 + 0.352i)5-s + (−0.529 − 0.171i)7-s + (0.336 − 0.109i)8-s − 0.423i·10-s + (0.932 − 0.360i)11-s + (1.11 + 1.52i)13-s + (0.121 + 0.374i)14-s + (−0.202 − 0.146i)16-s + (−0.223 + 0.307i)17-s + (0.784 − 0.255i)19-s + (−0.242 + 0.176i)20-s + (−0.593 − 0.383i)22-s + 1.31·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $0.981 + 0.189i$
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: \((2,\ 198,\ (\ :1),\ 0.981 + 0.189i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.34622 - 0.128474i\)
\(L(\frac12)\) \(\approx\) \(1.34622 - 0.128474i\)
\(L(2)\) 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 \( 1 + (-10.2 + 3.96i)T \)
good5 \( 1 + (-2.42 - 1.76i)T + (7.72 + 23.7i)T^{2} \)
7 \( 1 + (3.70 + 1.20i)T + (39.6 + 28.8i)T^{2} \)
13 \( 1 + (-14.4 - 19.8i)T + (-52.2 + 160. i)T^{2} \)
17 \( 1 + (3.79 - 5.22i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (-14.9 + 4.84i)T + (292. - 212. i)T^{2} \)
23 \( 1 - 30.3T + 529T^{2} \)
29 \( 1 + (-14.2 - 4.63i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (-21.4 + 15.5i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (1.29 - 3.97i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (41.2 - 13.3i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 - 56.0iT - 1.84e3T^{2} \)
47 \( 1 + (9.34 + 28.7i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-42.1 + 30.6i)T + (868. - 2.67e3i)T^{2} \)
59 \( 1 + (29.1 - 89.7i)T + (-2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (54.8 - 75.4i)T + (-1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 - 54.0T + 4.48e3T^{2} \)
71 \( 1 + (75.9 + 55.1i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (23.3 + 7.58i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (45.4 + 62.5i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-16.8 + 23.1i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 + 68.2T + 7.92e3T^{2} \)
97 \( 1 + (29.1 - 21.1i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−11.89241163797547414159198602082, −11.30859258022516689615571600760, −10.22086476322472455578082178690, −9.290014585421215323696558242601, −8.584327463768271679272539483026, −6.92857190916432211236297288332, −6.22704395959198910909986929729, −4.32665912029805933151935540931, −3.09210199171147930104393807392, −1.39028910168989221275783012987, 1.14464244695134852258049363373, 3.31848967887623789469324390092, 5.10108524346249990785355114922, 6.06984860932523380894029751189, 7.09493252867973976863704574286, 8.381585302634969102548027485066, 9.225580230416804771892644241622, 10.05955022879208834566119501335, 11.14584509956227597460712908433, 12.43571981905357032616109991608

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