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Properties

Label 2-198-11.5-c1-0-1
Degree $2$
Conductor $198$
Sign $-0.605 - 0.795i$
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  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−1.11 + 3.44i)5-s + (−0.5 + 0.363i)7-s + (−0.809 − 0.587i)8-s − 3.61·10-s + (−1.23 − 3.07i)11-s + (1 + 3.07i)13-s + (−0.5 − 0.363i)14-s + (0.309 − 0.951i)16-s + (−1.23 + 3.80i)17-s + (4.61 + 3.35i)19-s + (−1.11 − 3.44i)20-s + (2.54 − 2.12i)22-s + 5.70·23-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.404 + 0.293i)4-s + (−0.499 + 1.53i)5-s + (−0.188 + 0.137i)7-s + (−0.286 − 0.207i)8-s − 1.14·10-s + (−0.372 − 0.927i)11-s + (0.277 + 0.853i)13-s + (−0.133 − 0.0970i)14-s + (0.0772 − 0.237i)16-s + (−0.299 + 0.922i)17-s + (1.05 + 0.769i)19-s + (−0.249 − 0.769i)20-s + (0.542 − 0.453i)22-s + 1.19·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(1.58103\)
Root analytic conductor: \(1.25739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.482746 + 0.973759i\)
\(L(\frac12)\) \(\approx\) \(0.482746 + 0.973759i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 \)
11 \( 1 + (1.23 + 3.07i)T \)
good5 \( 1 + (1.11 - 3.44i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (0.5 - 0.363i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1 - 3.07i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.23 - 3.80i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-4.61 - 3.35i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 5.70T + 23T^{2} \)
29 \( 1 + (-5.54 + 4.02i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.04 + 3.21i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-7.23 + 5.25i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.61 + 3.35i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 4.76T + 43T^{2} \)
47 \( 1 + (-3.23 - 2.35i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.427 + 1.31i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-8.35 + 6.06i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.14 - 3.52i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 4.94T + 67T^{2} \)
71 \( 1 + (1.85 - 5.70i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (11.7 - 8.55i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.11 + 6.51i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.5 - 1.53i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 6.76T + 89T^{2} \)
97 \( 1 + (2.97 + 9.14i)T + (-78.4 + 57.0i)T^{2} \)
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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

−13.04477326286707957452147461931, −11.68236821443053849066771508300, −11.00739172285357545385274936939, −9.923566615626217658358838944845, −8.582607474740056543521135369136, −7.57699890059417488629179993851, −6.63397277956868486028479781620, −5.79133457360252736108124414702, −4.00606213643685081796472890640, −2.93990583611546244259795905837, 0.973237847268970108974317624707, 3.07555779999129346098457628821, 4.71677672887855939969065102470, 5.15677352242386663443661927017, 7.11213179247143156374948868445, 8.341752216856752443154052070060, 9.223548901742589469173585847518, 10.13897847365673060313615727897, 11.41869468664789614124433753058, 12.18848034855374806975238562521

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