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Properties

Label 2-198-9.4-c5-0-10
Degree $2$
Conductor $198$
Sign $-0.386 - 0.922i$
Analytic cond. $31.7559$
Root an. cond. $5.63524$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 3.46i)2-s + (−14.2 − 6.24i)3-s + (−7.99 + 13.8i)4-s + (−2.34 + 4.05i)5-s + (−6.93 − 61.9i)6-s + (−87.2 − 151. i)7-s − 63.9·8-s + (164. + 178. i)9-s − 18.7·10-s + (−60.5 − 104. i)11-s + (200. − 147. i)12-s + (177. − 307. i)13-s + (349. − 604. i)14-s + (58.7 − 43.3i)15-s + (−128 − 221. i)16-s + 324.·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.916 − 0.400i)3-s + (−0.249 + 0.433i)4-s + (−0.0418 + 0.0725i)5-s + (−0.0786 − 0.702i)6-s + (−0.673 − 1.16i)7-s − 0.353·8-s + (0.678 + 0.734i)9-s − 0.0592·10-s + (−0.150 − 0.261i)11-s + (0.402 − 0.296i)12-s + (0.291 − 0.504i)13-s + (0.476 − 0.824i)14-s + (0.0674 − 0.0496i)15-s + (−0.125 − 0.216i)16-s + 0.272·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $-0.386 - 0.922i$
Analytic conductor: \(31.7559\)
Root analytic conductor: \(5.63524\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :5/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7974010962\)
\(L(\frac12)\) \(\approx\) \(0.7974010962\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 - 3.46i)T \)
3 \( 1 + (14.2 + 6.24i)T \)
11 \( 1 + (60.5 + 104. i)T \)
good5 \( 1 + (2.34 - 4.05i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (87.2 + 151. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
13 \( 1 + (-177. + 307. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 - 324.T + 1.41e6T^{2} \)
19 \( 1 + 1.14e3T + 2.47e6T^{2} \)
23 \( 1 + (-536. + 930. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (-2.70e3 - 4.69e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (1.54e3 - 2.67e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 1.13e4T + 6.93e7T^{2} \)
41 \( 1 + (5.22e3 - 9.05e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (263. + 456. i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-7.48e3 - 1.29e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 - 1.00e4T + 4.18e8T^{2} \)
59 \( 1 + (2.44e4 - 4.22e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-2.38e4 - 4.13e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (5.87e3 - 1.01e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 408.T + 1.80e9T^{2} \)
73 \( 1 - 3.62e3T + 2.07e9T^{2} \)
79 \( 1 + (1.52e4 + 2.64e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (3.27e4 + 5.67e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 - 8.98e4T + 5.58e9T^{2} \)
97 \( 1 + (-3.66e4 - 6.35e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−12.12757749165515089537465978635, −10.79973279310847063247854489740, −10.29337739181841768811146718410, −8.686360832112676022147266021236, −7.38556327107446168492092388069, −6.78625173133345219976415697519, −5.75085324259887477680132353559, −4.60419402041332526876959410664, −3.30486737347651965755113911160, −1.03053944537681829954711738195, 0.29559250651027870492963409127, 2.11042177282775312481221995266, 3.61828352230109585686056796610, 4.84828603600944481210492321359, 5.85243033018199302618873425842, 6.71491811285442065675861482612, 8.621241815716224518432706816978, 9.576272761281827087457367201742, 10.40513013521957037785917116744, 11.44655417214386662093239926440

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