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Properties

Label 2-198-9.4-c1-0-5
Degree $2$
Conductor $198$
Sign $0.986 + 0.164i$
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  + (−0.5 − 0.866i)2-s + (1.72 − 0.158i)3-s + (−0.499 + 0.866i)4-s + (−0.724 + 1.25i)5-s + (−1 − 1.41i)6-s + (2.22 + 3.85i)7-s + 0.999·8-s + (2.94 − 0.548i)9-s + 1.44·10-s + (−0.5 − 0.866i)11-s + (−0.724 + 1.57i)12-s + (2.22 − 3.85i)13-s + (2.22 − 3.85i)14-s + (−1.05 + 2.28i)15-s + (−0.5 − 0.866i)16-s − 4.44·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.995 − 0.0917i)3-s + (−0.249 + 0.433i)4-s + (−0.324 + 0.561i)5-s + (−0.408 − 0.577i)6-s + (0.840 + 1.45i)7-s + 0.353·8-s + (0.983 − 0.182i)9-s + 0.458·10-s + (−0.150 − 0.261i)11-s + (−0.209 + 0.454i)12-s + (0.617 − 1.06i)13-s + (0.594 − 1.02i)14-s + (−0.271 + 0.588i)15-s + (−0.125 − 0.216i)16-s − 1.07·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $0.986 + 0.164i$
Analytic conductor: \(1.58103\)
Root analytic conductor: \(1.25739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :1/2),\ 0.986 + 0.164i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34119 - 0.111105i\)
\(L(\frac12)\) \(\approx\) \(1.34119 - 0.111105i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-1.72 + 0.158i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.724 - 1.25i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.22 - 3.85i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (-2.22 + 3.85i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.44T + 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.67 + 4.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.55T + 37T^{2} \)
41 \( 1 + (1.67 - 2.89i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.224 + 0.389i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.94 - 3.37i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 + (-0.275 + 0.476i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.89 + 11.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.17 + 3.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.79T + 71T^{2} \)
73 \( 1 + 13.3T + 73T^{2} \)
79 \( 1 + (2.67 + 4.63i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4 - 6.92i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 1.79T + 89T^{2} \)
97 \( 1 + (5.39 + 9.35i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.57735020830069214619231176550, −11.24736495827911753848213594862, −10.69858713182660500721382144153, −9.243105331325957475351188871279, −8.498789073565105285742797559222, −7.901051339778262455761669577752, −6.33661082736559964162577674806, −4.62125682339698817433234934355, −3.09173343881339742077211369632, −2.13517902863921683897429506782, 1.61163280136181449351656857419, 4.06588353658832709183778938010, 4.64205554294573931131343200866, 6.74152623426404265738314088047, 7.53537519926816667846959414845, 8.499165044944569854238952675336, 9.155648018254670638613224164324, 10.45400710099286411943541459745, 11.23857805271348040380662966360, 12.93731066074545700242887584215

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