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Properties

Label 2-198-33.14-c2-0-4
Degree $2$
Conductor $198$
Sign $-0.217 + 0.976i$
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 + (−2.92 + 4.02i)5-s + (2.20 − 6.78i)7-s + (2.68 − 0.874i)8-s + 7.02·10-s + (−9.66 − 5.24i)11-s + (17.3 − 12.5i)13-s + (−9.60 + 3.11i)14-s + (−3.23 − 2.35i)16-s + (15.6 − 21.5i)17-s + (−3.99 − 12.2i)19-s + (−5.84 − 8.04i)20-s + (2.03 + 15.4i)22-s − 4.58i·23-s + ⋯
L(s)  = 1  + (−0.415 − 0.572i)2-s + (−0.154 + 0.475i)4-s + (−0.584 + 0.804i)5-s + (0.315 − 0.969i)7-s + (0.336 − 0.109i)8-s + 0.702·10-s + (−0.879 − 0.476i)11-s + (1.33 − 0.967i)13-s + (−0.685 + 0.222i)14-s + (−0.202 − 0.146i)16-s + (0.922 − 1.26i)17-s + (−0.210 − 0.646i)19-s + (−0.292 − 0.402i)20-s + (0.0926 + 0.701i)22-s − 0.199i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 + 0.976i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.217 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $-0.217 + 0.976i$
Analytic conductor: \(5.39510\)
Root analytic conductor: \(2.32273\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :1),\ -0.217 + 0.976i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.600552 - 0.749235i\)
\(L(\frac12)\) \(\approx\) \(0.600552 - 0.749235i\)
\(L(2)\) 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 \( 1 + (9.66 + 5.24i)T \)
good5 \( 1 + (2.92 - 4.02i)T + (-7.72 - 23.7i)T^{2} \)
7 \( 1 + (-2.20 + 6.78i)T + (-39.6 - 28.8i)T^{2} \)
13 \( 1 + (-17.3 + 12.5i)T + (52.2 - 160. i)T^{2} \)
17 \( 1 + (-15.6 + 21.5i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (3.99 + 12.2i)T + (-292. + 212. i)T^{2} \)
23 \( 1 + 4.58iT - 529T^{2} \)
29 \( 1 + (33.2 + 10.7i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (19.2 - 13.9i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (-16.0 + 49.3i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (-30.2 + 9.82i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 + 32.0T + 1.84e3T^{2} \)
47 \( 1 + (-30.5 + 9.91i)T + (1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (-28.0 - 38.6i)T + (-868. + 2.67e3i)T^{2} \)
59 \( 1 + (-21.3 - 6.93i)T + (2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (-26.7 - 19.4i)T + (1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + 109.T + 4.48e3T^{2} \)
71 \( 1 + (0.836 - 1.15i)T + (-1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (42.7 - 131. i)T + (-4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (-44.9 + 32.6i)T + (1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (-94.8 + 130. i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 - 31.8iT - 7.92e3T^{2} \)
97 \( 1 + (-105. + 76.6i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−11.59541882482515358102711368302, −10.84016599856608953025187296225, −10.44934840730535304754644248435, −9.011257204287142381085725593115, −7.76608820513194273389623683385, −7.28646311770203875713710628424, −5.55426225953537601857654510416, −3.88099636950881234189511305513, −2.91817110038437092985339235081, −0.67519066965641040257986650499, 1.66114020643892750966011111990, 3.95934768453267323787061246708, 5.28074980568643313744539004194, 6.20773282505439419004063609306, 7.80043319391670910241451088261, 8.404589653858823163572120243865, 9.239468053711752661012964970296, 10.48907147748105931044623191835, 11.60576139174738547681435811421, 12.50965242991597922013367623661

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