↑

Properties

Label 2-198-33.17-c1-0-0
Degree $2$
Conductor $198$
Sign $-0.0214 - 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−1.05 − 0.343i)5-s + (2.97 + 4.09i)7-s + (−0.809 − 0.587i)8-s − 1.11i·10-s + (−0.0444 + 3.31i)11-s + (−0.0598 + 0.0194i)13-s + (−2.97 + 4.09i)14-s + (0.309 − 0.951i)16-s + (−0.486 + 1.49i)17-s + (2.74 − 3.78i)19-s + (1.05 − 0.343i)20-s + (−3.16 + 0.982i)22-s − 7.56i·23-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.404 + 0.293i)4-s + (−0.473 − 0.153i)5-s + (1.12 + 1.54i)7-s + (−0.286 − 0.207i)8-s − 0.351i·10-s + (−0.0134 + 0.999i)11-s + (−0.0165 + 0.00539i)13-s + (−0.795 + 1.09i)14-s + (0.0772 − 0.237i)16-s + (−0.117 + 0.362i)17-s + (0.630 − 0.868i)19-s + (0.236 − 0.0768i)20-s + (−0.675 + 0.209i)22-s − 1.57i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0214 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0214 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $-0.0214 - 0.999i$
Analytic conductor: \(1.58103\)
Root analytic conductor: \(1.25739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :1/2),\ -0.0214 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.884706 + 0.903857i\)
\(L(\frac12)\) \(\approx\) \(0.884706 + 0.903857i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0444 - 3.31i)T \)
good5 \( 1 + (1.05 + 0.343i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (-2.97 - 4.09i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.0598 - 0.0194i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.486 - 1.49i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.74 + 3.78i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 7.56iT - 23T^{2} \)
29 \( 1 + (-2.03 + 1.47i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.789 - 2.42i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.513 + 0.373i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.87 - 4.99i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 9.52iT - 43T^{2} \)
47 \( 1 + (-0.490 + 0.675i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-5.23 + 1.69i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.96 + 8.20i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (12.2 + 3.97i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 2.91T + 67T^{2} \)
71 \( 1 + (-10.5 - 3.43i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.19 + 5.77i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-10.7 + 3.49i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.48 - 10.7i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 11.4iT - 89T^{2} \)
97 \( 1 + (-1.30 - 4.03i)T + (-78.4 + 57.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.42532761509837414605345718684, −12.11768739678475567170698332189, −10.95715091840848164704204436056, −9.444814555969975854782948209048, −8.517232359674923257648974910160, −7.79643715376505998823217837188, −6.48483362689578796434528717584, −5.21770502743655972690647312270, −4.43644573944028933885516935960, −2.38858336267096770743129472010, 1.23533685471533350310884926587, 3.41422456535717678874805259078, 4.34445362891450090655216061699, 5.66122847317248441553245400727, 7.40503249435694379977477942481, 8.027558072584218302886102699811, 9.475180610883994806937702384231, 10.61939261683360196978107121162, 11.24231246585385342645013191791, 11.92200895622162702867377771240

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