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Properties

Label 2-198-11.4-c3-0-2
Degree $2$
Conductor $198$
Sign $0.0776 - 0.996i$
Analytic cond. $11.6823$
Root an. cond. $3.41794$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.61 − 1.17i)2-s + (1.23 + 3.80i)4-s + (−6.52 + 4.73i)5-s + (−8.05 − 24.8i)7-s + (2.47 − 7.60i)8-s + 16.1·10-s + (33.3 + 14.7i)11-s + (2.64 + 1.91i)13-s + (−16.1 + 49.6i)14-s + (−12.9 + 9.40i)16-s + (−16.8 + 12.2i)17-s + (−38.9 + 119. i)19-s + (−26.0 − 18.9i)20-s + (−36.6 − 63.0i)22-s − 97.8·23-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.154 + 0.475i)4-s + (−0.583 + 0.423i)5-s + (−0.435 − 1.33i)7-s + (0.109 − 0.336i)8-s + 0.509·10-s + (0.914 + 0.403i)11-s + (0.0563 + 0.0409i)13-s + (−0.307 + 0.946i)14-s + (−0.202 + 0.146i)16-s + (−0.240 + 0.175i)17-s + (−0.470 + 1.44i)19-s + (−0.291 − 0.211i)20-s + (−0.355 − 0.611i)22-s − 0.886·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0776 - 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0776 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $0.0776 - 0.996i$
Analytic conductor: \(11.6823\)
Root analytic conductor: \(3.41794\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :3/2),\ 0.0776 - 0.996i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.448592 + 0.415014i\)
\(L(\frac12)\) \(\approx\) \(0.448592 + 0.415014i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.61 + 1.17i)T \)
3 \( 1 \)
11 \( 1 + (-33.3 - 14.7i)T \)
good5 \( 1 + (6.52 - 4.73i)T + (38.6 - 118. i)T^{2} \)
7 \( 1 + (8.05 + 24.8i)T + (-277. + 201. i)T^{2} \)
13 \( 1 + (-2.64 - 1.91i)T + (678. + 2.08e3i)T^{2} \)
17 \( 1 + (16.8 - 12.2i)T + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (38.9 - 119. i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 + 97.8T + 1.21e4T^{2} \)
29 \( 1 + (-81.5 - 250. i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (161. + 117. i)T + (9.20e3 + 2.83e4i)T^{2} \)
37 \( 1 + (-112. - 347. i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (84.5 - 260. i)T + (-5.57e4 - 4.05e4i)T^{2} \)
43 \( 1 - 388.T + 7.95e4T^{2} \)
47 \( 1 + (-16.0 + 49.3i)T + (-8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (333. + 242. i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (8.12 + 24.9i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (132. - 96.4i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 - 276.T + 3.00e5T^{2} \)
71 \( 1 + (418. - 303. i)T + (1.10e5 - 3.40e5i)T^{2} \)
73 \( 1 + (74.6 + 229. i)T + (-3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (-220. - 160. i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (-58.7 + 42.6i)T + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 - 1.19e3T + 7.04e5T^{2} \)
97 \( 1 + (1.18e3 + 860. i)T + (2.82e5 + 8.68e5i)T^{2} \)
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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

−12.09981559892913302943558556363, −11.12353196167879250628041380818, −10.32362555368154980182855210902, −9.511944839735030193183430422451, −8.166517287043383655491370985549, −7.27506584868782960639808855530, −6.38419598411137438494672502166, −4.20960322749867423597688744890, −3.46049062598469595599816983887, −1.43510482163779866977397970884, 0.33461357162972588458734610786, 2.39928628298257780809585660593, 4.20810190088723532540820292354, 5.68435850143458572787790284752, 6.57177329397393113816270717122, 7.905335116891865283576802037428, 8.950680086128526421964233809683, 9.309127819588274231675046535898, 10.85135934874705867569247350025, 11.83041585169280559058942604326

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