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Properties

Label 2-198-33.17-c1-0-2
Degree $2$
Conductor $198$
Sign $0.437 - 0.899i$
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 + (3.29 + 1.07i)5-s + (−0.740 − 1.01i)7-s + (−0.809 − 0.587i)8-s + 3.46i·10-s + (3.28 + 0.487i)11-s + (−4.41 + 1.43i)13-s + (0.740 − 1.01i)14-s + (0.309 − 0.951i)16-s + (−1.51 + 4.65i)17-s + (1.72 − 2.37i)19-s + (−3.29 + 1.07i)20-s + (0.549 + 3.27i)22-s − 4.74i·23-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.404 + 0.293i)4-s + (1.47 + 0.478i)5-s + (−0.279 − 0.385i)7-s + (−0.286 − 0.207i)8-s + 1.09i·10-s + (0.989 + 0.147i)11-s + (−1.22 + 0.397i)13-s + (0.197 − 0.272i)14-s + (0.0772 − 0.237i)16-s + (−0.367 + 1.12i)17-s + (0.395 − 0.543i)19-s + (−0.736 + 0.239i)20-s + (0.117 + 0.697i)22-s − 0.988i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $0.437 - 0.899i$
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: \((2,\ 198,\ (\ :1/2),\ 0.437 - 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26885 + 0.793865i\)
\(L(\frac12)\) \(\approx\) \(1.26885 + 0.793865i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.28 - 0.487i)T \)
good5 \( 1 + (-3.29 - 1.07i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.740 + 1.01i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (4.41 - 1.43i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.51 - 4.65i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.72 + 2.37i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 4.74iT - 23T^{2} \)
29 \( 1 + (0.654 - 0.475i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.17 + 9.75i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.513 - 0.373i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (8.87 + 6.44i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 2.78iT - 43T^{2} \)
47 \( 1 + (2.19 - 3.02i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-5.62 + 1.82i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-5.43 - 7.47i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-10.5 - 3.42i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 + (-5.59 - 1.81i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.65 + 2.28i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (15.7 - 5.11i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.480 + 1.47i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 11.4iT - 89T^{2} \)
97 \( 1 + (-4.63 - 14.2i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.99607709248735891256641375022, −11.82511527212343727112586535621, −10.39757523593594081533276442368, −9.676971471242899051800454819939, −8.780720791817693313470182921284, −7.13981755245189718152551420056, −6.52094190177658712898028159699, −5.47916217318311199733899348776, −4.06937684373197097816460644742, −2.24023806054210922617418734513, 1.66107112240000205828567178922, 3.06421135647933448434080744521, 4.91640447436882069772226375689, 5.69958007443165578070768635970, 6.97962582301375580871975473871, 8.770060197584423198399829514688, 9.587156146496815729701750385821, 10.05625271056784499877336300793, 11.53086798107670116037539106492, 12.32563863396116703458402977125

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