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Properties

Label 2-198-11.2-c2-0-9
Degree $2$
Conductor $198$
Sign $0.642 + 0.766i$
Analytic cond. $5.39510$
Root an. cond. $2.32273$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.34 + 0.437i)2-s + (1.61 + 1.17i)4-s + (−1.96 − 6.03i)5-s + (5.91 − 8.14i)7-s + (1.66 + 2.28i)8-s − 8.97i·10-s + (−10.7 − 2.43i)11-s + (−1.07 − 0.348i)13-s + (11.5 − 8.36i)14-s + (1.23 + 3.80i)16-s + (18.8 − 6.12i)17-s + (0.867 + 1.19i)19-s + (3.92 − 12.0i)20-s + (−13.3 − 7.96i)22-s + 38.3·23-s + ⋯
L(s)  = 1  + (0.672 + 0.218i)2-s + (0.404 + 0.293i)4-s + (−0.392 − 1.20i)5-s + (0.845 − 1.16i)7-s + (0.207 + 0.286i)8-s − 0.897i·10-s + (−0.975 − 0.221i)11-s + (−0.0823 − 0.0267i)13-s + (0.822 − 0.597i)14-s + (0.0772 + 0.237i)16-s + (1.10 − 0.360i)17-s + (0.0456 + 0.0628i)19-s + (0.196 − 0.603i)20-s + (−0.607 − 0.361i)22-s + 1.66·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(5.39510\)
Root analytic conductor: \(2.32273\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :1),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.95346 - 0.911976i\)
\(L(\frac12)\) \(\approx\) \(1.95346 - 0.911976i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 - 0.437i)T \)
3 \( 1 \)
11 \( 1 + (10.7 + 2.43i)T \)
good5 \( 1 + (1.96 + 6.03i)T + (-20.2 + 14.6i)T^{2} \)
7 \( 1 + (-5.91 + 8.14i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (1.07 + 0.348i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (-18.8 + 6.12i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-0.867 - 1.19i)T + (-111. + 343. i)T^{2} \)
23 \( 1 - 38.3T + 529T^{2} \)
29 \( 1 + (10.1 - 13.9i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (18.5 - 57.1i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (19.5 + 14.2i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-19.6 - 27.0i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 42.3iT - 1.84e3T^{2} \)
47 \( 1 + (48.3 - 35.0i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (-7.40 + 22.7i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-35.0 - 25.4i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (97.0 - 31.5i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 - 110.T + 4.48e3T^{2} \)
71 \( 1 + (10.2 + 31.6i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (26.8 - 37.0i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (-73.0 - 23.7i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-67.1 + 21.8i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + 53.1T + 7.92e3T^{2} \)
97 \( 1 + (-33.4 + 102. i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−12.38898684550566059387861573942, −11.22502209647012369110303027267, −10.42187163417204185034447087643, −8.870728901922722364456382663261, −7.88255243750401289020452063931, −7.15307245389746463202049013662, −5.24501607128308132373166754354, −4.81472047213313966370050446939, −3.41080074740274182240111017850, −1.12388018892533822105479584960, 2.25925265337902756113178719022, 3.31283967150493241336836707833, 4.96380178319144670731980347976, 5.89177151010981407302365733931, 7.25837554728198854135993631061, 8.119289252975768953860359833283, 9.624912555644121260591345703441, 10.83486606748005149404066388384, 11.35983699783631489933595029543, 12.31805268189921137311723146556

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