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Properties

Label 2-198-11.5-c3-0-0
Degree $2$
Conductor $198$
Sign $-0.619 + 0.785i$
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  + (0.618 + 1.90i)2-s + (−3.23 + 2.35i)4-s + (−4.60 + 14.1i)5-s + (−17.6 + 12.7i)7-s + (−6.47 − 4.70i)8-s − 29.8·10-s + (29.3 − 21.6i)11-s + (−13.6 − 41.8i)13-s + (−35.2 − 25.5i)14-s + (4.94 − 15.2i)16-s + (7.69 − 23.6i)17-s + (−17.7 − 12.9i)19-s + (−18.4 − 56.7i)20-s + (59.3 + 42.5i)22-s − 177.·23-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.404 + 0.293i)4-s + (−0.412 + 1.26i)5-s + (−0.950 + 0.690i)7-s + (−0.286 − 0.207i)8-s − 0.943·10-s + (0.805 − 0.592i)11-s + (−0.290 − 0.893i)13-s + (−0.672 − 0.488i)14-s + (0.0772 − 0.237i)16-s + (0.109 − 0.338i)17-s + (−0.214 − 0.155i)19-s + (−0.206 − 0.634i)20-s + (0.574 + 0.411i)22-s − 1.61·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 + 0.785i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.619 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $-0.619 + 0.785i$
Analytic conductor: \(11.6823\)
Root analytic conductor: \(3.41794\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :3/2),\ -0.619 + 0.785i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.211894 - 0.436894i\)
\(L(\frac12)\) \(\approx\) \(0.211894 - 0.436894i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.618 - 1.90i)T \)
3 \( 1 \)
11 \( 1 + (-29.3 + 21.6i)T \)
good5 \( 1 + (4.60 - 14.1i)T + (-101. - 73.4i)T^{2} \)
7 \( 1 + (17.6 - 12.7i)T + (105. - 326. i)T^{2} \)
13 \( 1 + (13.6 + 41.8i)T + (-1.77e3 + 1.29e3i)T^{2} \)
17 \( 1 + (-7.69 + 23.6i)T + (-3.97e3 - 2.88e3i)T^{2} \)
19 \( 1 + (17.7 + 12.9i)T + (2.11e3 + 6.52e3i)T^{2} \)
23 \( 1 + 177.T + 1.21e4T^{2} \)
29 \( 1 + (120. - 87.8i)T + (7.53e3 - 2.31e4i)T^{2} \)
31 \( 1 + (-23.2 - 71.4i)T + (-2.41e4 + 1.75e4i)T^{2} \)
37 \( 1 + (179. - 130. i)T + (1.56e4 - 4.81e4i)T^{2} \)
41 \( 1 + (204. + 148. i)T + (2.12e4 + 6.55e4i)T^{2} \)
43 \( 1 - 130.T + 7.95e4T^{2} \)
47 \( 1 + (-403. - 293. i)T + (3.20e4 + 9.87e4i)T^{2} \)
53 \( 1 + (3.99 + 12.3i)T + (-1.20e5 + 8.75e4i)T^{2} \)
59 \( 1 + (-28.7 + 20.9i)T + (6.34e4 - 1.95e5i)T^{2} \)
61 \( 1 + (-166. + 511. i)T + (-1.83e5 - 1.33e5i)T^{2} \)
67 \( 1 + 519.T + 3.00e5T^{2} \)
71 \( 1 + (24.2 - 74.6i)T + (-2.89e5 - 2.10e5i)T^{2} \)
73 \( 1 + (925. - 672. i)T + (1.20e5 - 3.69e5i)T^{2} \)
79 \( 1 + (-238. - 734. i)T + (-3.98e5 + 2.89e5i)T^{2} \)
83 \( 1 + (166. - 510. i)T + (-4.62e5 - 3.36e5i)T^{2} \)
89 \( 1 + 667.T + 7.04e5T^{2} \)
97 \( 1 + (55.5 + 170. i)T + (-7.38e5 + 5.36e5i)T^{2} \)
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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

−12.62324839767261180077889307246, −11.82862156538856553576993750099, −10.66273167674280367701049579962, −9.644707922176861690897238540690, −8.524040183239884677077865233313, −7.34460063211585816940999712975, −6.47064186618915467565474912012, −5.62893866709388028107421168544, −3.76554483297691817402730714377, −2.85762738684233419360763824539, 0.19172092051388117392342167545, 1.75699250519106437388691991434, 3.85845056805569284350356103262, 4.40571185273252384607696928085, 5.97497720206440380454113877460, 7.29088030669718543633910036150, 8.658591023873469567094549896600, 9.518818750045292118305179778686, 10.29146791280967414276193351119, 11.80320558378577034627035159234

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