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Properties

Label 2-198-9.7-c5-0-18
Degree $2$
Conductor $198$
Sign $0.0423 - 0.999i$
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 + (8.66 + 12.9i)3-s + (−7.99 − 13.8i)4-s + (51.5 + 89.2i)5-s + (62.2 − 4.11i)6-s + (17.1 − 29.6i)7-s − 63.9·8-s + (−92.7 + 224. i)9-s + 412.·10-s + (−60.5 + 104. i)11-s + (110. − 223. i)12-s + (158. + 275. i)13-s + (−68.5 − 118. i)14-s + (−709. + 1.44e3i)15-s + (−128 + 221. i)16-s + 1.65e3·17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.556 + 0.831i)3-s + (−0.249 − 0.433i)4-s + (0.921 + 1.59i)5-s + (0.705 − 0.0466i)6-s + (0.132 − 0.228i)7-s − 0.353·8-s + (−0.381 + 0.924i)9-s + 1.30·10-s + (−0.150 + 0.261i)11-s + (0.220 − 0.448i)12-s + (0.260 + 0.451i)13-s + (−0.0934 − 0.161i)14-s + (−0.814 + 1.65i)15-s + (−0.125 + 0.216i)16-s + 1.38·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0423 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0423 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $0.0423 - 0.999i$
Analytic conductor: \(31.7559\)
Root analytic conductor: \(5.63524\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :5/2),\ 0.0423 - 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.098478875\)
\(L(\frac12)\) \(\approx\) \(3.098478875\)
\(L(\frac{7}{2})\) 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 \( 1 + (-8.66 - 12.9i)T \)
11 \( 1 + (60.5 - 104. i)T \)
good5 \( 1 + (-51.5 - 89.2i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-17.1 + 29.6i)T + (-8.40e3 - 1.45e4i)T^{2} \)
13 \( 1 + (-158. - 275. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 1.65e3T + 1.41e6T^{2} \)
19 \( 1 + 431.T + 2.47e6T^{2} \)
23 \( 1 + (1.55e3 + 2.69e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (1.66e3 - 2.89e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (2.33e3 + 4.04e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + 7.83e3T + 6.93e7T^{2} \)
41 \( 1 + (-6.99e3 - 1.21e4i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (7.49e3 - 1.29e4i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (-2.02e3 + 3.49e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 - 3.08e4T + 4.18e8T^{2} \)
59 \( 1 + (1.28e4 + 2.23e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-2.37e4 + 4.11e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.84e4 - 3.19e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 2.49e4T + 1.80e9T^{2} \)
73 \( 1 - 4.38e4T + 2.07e9T^{2} \)
79 \( 1 + (1.81e3 - 3.14e3i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-4.29e3 + 7.43e3i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 7.00e4T + 5.58e9T^{2} \)
97 \( 1 + (-3.93e4 + 6.82e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−11.50954101597589939974634750763, −10.62313659138127305689448889277, −10.12219394099616302733761867509, −9.323151704465365169998721777694, −7.83117550513730411305785539159, −6.49841512421778830605754173919, −5.36996926559918996994694566965, −3.91270819321974168820511183251, −2.93218312143380892611874162012, −1.94185308593967583564368463199, 0.77294898293199175365342908299, 1.97879018951304024704621543430, 3.69899001712301673743172829323, 5.42507401964016836832589802729, 5.79325914180753672031854997521, 7.36877715295732233967927108513, 8.376710437414313593964696881253, 8.949821487183815130908756822646, 10.02131085022849334411276471028, 12.03412455284291787692701325574

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