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Properties

Label 2-198-33.8-c1-0-3
Degree $2$
Conductor $198$
Sign $0.318 + 0.947i$
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 + (1.63 − 2.24i)5-s + (−4.12 − 1.34i)7-s + (−0.309 − 0.951i)8-s − 2.77i·10-s + (3.30 + 0.238i)11-s + (2.74 + 3.78i)13-s + (−4.12 + 1.34i)14-s + (−0.809 − 0.587i)16-s + (3.17 + 2.30i)17-s + (−4.41 + 1.43i)19-s + (−1.63 − 2.24i)20-s + (2.81 − 1.75i)22-s − 2.86i·23-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (0.729 − 1.00i)5-s + (−1.55 − 0.506i)7-s + (−0.109 − 0.336i)8-s − 0.877i·10-s + (0.997 + 0.0719i)11-s + (0.762 + 1.04i)13-s + (−1.10 + 0.358i)14-s + (−0.202 − 0.146i)16-s + (0.770 + 0.559i)17-s + (−1.01 + 0.328i)19-s + (−0.364 − 0.502i)20-s + (0.600 − 0.373i)22-s − 0.597i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $0.318 + 0.947i$
Analytic conductor: \(1.58103\)
Root analytic conductor: \(1.25739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :1/2),\ 0.318 + 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30341 - 0.936710i\)
\(L(\frac12)\) \(\approx\) \(1.30341 - 0.936710i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.30 - 0.238i)T \)
good5 \( 1 + (-1.63 + 2.24i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (4.12 + 1.34i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-2.74 - 3.78i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.17 - 2.30i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.41 - 1.43i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.86iT - 23T^{2} \)
29 \( 1 + (0.977 - 3.00i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.90 + 2.83i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.17 - 6.69i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.290 - 0.892i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.98iT - 43T^{2} \)
47 \( 1 + (5.02 - 1.63i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (7.77 + 10.6i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-4.93 - 1.60i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-5.46 + 7.51i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 2.82T + 67T^{2} \)
71 \( 1 + (0.944 - 1.30i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (13.6 + 4.43i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.11 + 7.04i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-7.71 - 5.60i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 7.05iT - 89T^{2} \)
97 \( 1 + (8.66 - 6.29i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.62773395226731779789862000111, −11.50791156440250273648933058161, −10.16103047380634183656032509674, −9.530999315825299323047042832116, −8.593131595571291318004539553515, −6.57547760963847169259658372974, −6.14119091906624076268649507144, −4.52373075614418280205280526141, −3.48700913567417690088334077102, −1.48919953620733065135636987581, 2.74719216616018268915642164748, 3.66386119770901267454302900556, 5.71841089035157705925766903064, 6.29120643649505583371443956412, 7.15011321719857237619087085618, 8.747746463886811139309369572921, 9.768553632677473957398932940886, 10.64352033353989070039709721531, 11.92434781459927068098165318279, 12.86639506256210397703660943667

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