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Properties

Label 2-198-33.29-c1-0-1
Degree $2$
Conductor $198$
Sign $0.667 - 0.744i$
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  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.604 + 0.831i)5-s + (1.88 − 0.613i)7-s + (−0.309 + 0.951i)8-s + 1.02i·10-s + (−2.07 + 2.58i)11-s + (1.72 − 2.37i)13-s + (1.88 + 0.613i)14-s + (−0.809 + 0.587i)16-s + (−1.17 + 0.854i)17-s + (−0.0598 − 0.0194i)19-s + (−0.604 + 0.831i)20-s + (−3.19 + 0.877i)22-s + 0.0388i·23-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.154 + 0.475i)4-s + (0.270 + 0.371i)5-s + (0.714 − 0.232i)7-s + (−0.109 + 0.336i)8-s + 0.325i·10-s + (−0.624 + 0.780i)11-s + (0.477 − 0.657i)13-s + (0.504 + 0.164i)14-s + (−0.202 + 0.146i)16-s + (−0.285 + 0.207i)17-s + (−0.0137 − 0.00445i)19-s + (−0.135 + 0.185i)20-s + (−0.681 + 0.187i)22-s + 0.00810i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $0.667 - 0.744i$
Analytic conductor: \(1.58103\)
Root analytic conductor: \(1.25739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :1/2),\ 0.667 - 0.744i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53724 + 0.685875i\)
\(L(\frac12)\) \(\approx\) \(1.53724 + 0.685875i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 \)
11 \( 1 + (2.07 - 2.58i)T \)
good5 \( 1 + (-0.604 - 0.831i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.88 + 0.613i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.72 + 2.37i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.17 - 0.854i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.0598 + 0.0194i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.0388iT - 23T^{2} \)
29 \( 1 + (2.64 + 8.12i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (8.52 + 6.19i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.17 - 6.69i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.70 + 5.26i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 5.07iT - 43T^{2} \)
47 \( 1 + (6.68 + 2.17i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.62 + 4.98i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-4.53 + 1.47i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.24 - 8.59i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 15.6T + 67T^{2} \)
71 \( 1 + (-7.12 - 9.80i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-14.5 + 4.71i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-0.115 + 0.158i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.71 - 3.42i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 7.05iT - 89T^{2} \)
97 \( 1 + (3.28 + 2.38i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.90911575214340179597905528140, −11.66024079600968141612864603399, −10.77719470431833458175460516879, −9.763758975697671792192683714147, −8.261693010724992257565920304832, −7.51433466173984435899828831764, −6.29151761955871695669674104204, −5.19660140346821557239020930699, −4.01301184209862972249549011813, −2.33205850929931235825171678174, 1.73196071109335370564916458252, 3.41646416191963056771288697803, 4.90619474880810163179641925714, 5.69799976671410657620892444263, 7.11724851920848905252730016853, 8.531292173979407265241456758603, 9.326966950134310303935898041775, 10.85170711193777264922527425159, 11.21957498325855061369909160075, 12.47852433229344648915709475122

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