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Properties

Label 2-198-11.2-c2-0-6
Degree $2$
Conductor $198$
Sign $0.591 - 0.806i$
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 + (1.98 + 6.12i)5-s + (4.81 − 6.63i)7-s + (1.66 + 2.28i)8-s + 9.10i·10-s + (0.195 + 10.9i)11-s + (−8.81 − 2.86i)13-s + (9.37 − 6.81i)14-s + (1.23 + 3.80i)16-s + (3.81 − 1.24i)17-s + (16.9 + 23.3i)19-s + (−3.97 + 12.2i)20-s + (−4.54 + 14.8i)22-s − 5.74·23-s + ⋯
L(s)  = 1  + (0.672 + 0.218i)2-s + (0.404 + 0.293i)4-s + (0.397 + 1.22i)5-s + (0.688 − 0.947i)7-s + (0.207 + 0.286i)8-s + 0.910i·10-s + (0.0177 + 0.999i)11-s + (−0.678 − 0.220i)13-s + (0.669 − 0.486i)14-s + (0.0772 + 0.237i)16-s + (0.224 − 0.0729i)17-s + (0.894 + 1.23i)19-s + (−0.198 + 0.612i)20-s + (−0.206 + 0.676i)22-s − 0.249·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $0.591 - 0.806i$
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: \((2,\ 198,\ (\ :1),\ 0.591 - 0.806i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.19060 + 1.11035i\)
\(L(\frac12)\) \(\approx\) \(2.19060 + 1.11035i\)
\(L(2)\) 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 \( 1 + (-0.195 - 10.9i)T \)
good5 \( 1 + (-1.98 - 6.12i)T + (-20.2 + 14.6i)T^{2} \)
7 \( 1 + (-4.81 + 6.63i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (8.81 + 2.86i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (-3.81 + 1.24i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-16.9 - 23.3i)T + (-111. + 343. i)T^{2} \)
23 \( 1 + 5.74T + 529T^{2} \)
29 \( 1 + (-17.8 + 24.5i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-16.3 + 50.4i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (41.2 + 29.9i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (44.9 + 61.9i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 9.29iT - 1.84e3T^{2} \)
47 \( 1 + (61.9 - 45.0i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (1.63 - 5.04i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (31.5 + 22.9i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-76.7 + 24.9i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + 7.38T + 4.48e3T^{2} \)
71 \( 1 + (34.2 + 105. i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (16.0 - 22.0i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (-41.7 - 13.5i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-13.2 + 4.30i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 58.6T + 7.92e3T^{2} \)
97 \( 1 + (-50.0 + 154. i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−12.38232792066531041730832116431, −11.51007549156951368914730111264, −10.38525710708145921370058717420, −9.874582314943878511332418551214, −7.81203133359677868383763109957, −7.28933133980513005724201880045, −6.17526855481370416816157220563, −4.85640018970569710190278644968, −3.61414492750881039616292682339, −2.11220051463519650064764775306, 1.39949619979340838762736083441, 3.02156470128311486238589775909, 5.06326815462827513278147518966, 5.13438869425292434991685140323, 6.68315207817996879721643267422, 8.353740549442822442139890548210, 8.966203239979594614512858160377, 10.18557589006004689308124305116, 11.60130678216540997395630620416, 12.01542776199634866483596491471

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