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Properties

Label 2-198-9.4-c5-0-28
Degree $2$
Conductor $198$
Sign $-0.354 - 0.934i$
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 + (14.1 + 6.49i)3-s + (−7.99 + 13.8i)4-s + (23.5 − 40.8i)5-s + (5.86 + 62.0i)6-s + (78.0 + 135. i)7-s − 63.9·8-s + (158. + 183. i)9-s + 188.·10-s + (−60.5 − 104. i)11-s + (−203. + 144. i)12-s + (−289. + 502. i)13-s + (−312. + 540. i)14-s + (599. − 425. i)15-s + (−128 − 221. i)16-s + 1.71e3·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.909 + 0.416i)3-s + (−0.249 + 0.433i)4-s + (0.421 − 0.730i)5-s + (0.0664 + 0.703i)6-s + (0.602 + 1.04i)7-s − 0.353·8-s + (0.653 + 0.757i)9-s + 0.596·10-s + (−0.150 − 0.261i)11-s + (−0.407 + 0.289i)12-s + (−0.475 + 0.824i)13-s + (−0.425 + 0.737i)14-s + (0.687 − 0.488i)15-s + (−0.125 − 0.216i)16-s + 1.43·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 - 0.934i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.354 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $-0.354 - 0.934i$
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: \((2,\ 198,\ (\ :5/2),\ -0.354 - 0.934i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.628287068\)
\(L(\frac12)\) \(\approx\) \(3.628287068\)
\(L(\frac{7}{2})\) 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 \( 1 + (-14.1 - 6.49i)T \)
11 \( 1 + (60.5 + 104. i)T \)
good5 \( 1 + (-23.5 + 40.8i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-78.0 - 135. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
13 \( 1 + (289. - 502. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 - 1.71e3T + 1.41e6T^{2} \)
19 \( 1 + 2.11e3T + 2.47e6T^{2} \)
23 \( 1 + (-1.74e3 + 3.02e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (-3.91e3 - 6.77e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (3.21e3 - 5.56e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 6.04e3T + 6.93e7T^{2} \)
41 \( 1 + (2.51e3 - 4.35e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (-1.18e3 - 2.04e3i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (1.01e4 + 1.75e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + 2.87e4T + 4.18e8T^{2} \)
59 \( 1 + (8.65e3 - 1.49e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.51e4 + 2.62e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-2.51e4 + 4.36e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 6.67e4T + 1.80e9T^{2} \)
73 \( 1 + 3.11e4T + 2.07e9T^{2} \)
79 \( 1 + (-2.76e4 - 4.78e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-1.28e3 - 2.21e3i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + 1.20e5T + 5.58e9T^{2} \)
97 \( 1 + (-5.04e4 - 8.73e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−12.37286928458759717718841961113, −10.83351912910752868789963984829, −9.522146808104877029431024180243, −8.709779370009602436891618145884, −8.194027600948347878926656286307, −6.74786004085184150486982653062, −5.26555367114016532753441874076, −4.67708961948277938838774396177, −3.07107350038641584714458548844, −1.72946904705678762938556631401, 0.920806391180619662843605787552, 2.23167539803902223302647369731, 3.29851227259707557638696831560, 4.48557873817833043808522430825, 6.08042917227779295486579557900, 7.39503953713058123581656572031, 8.067775974064928112269570790516, 9.670746554325751256661173350509, 10.24270872465904155045524244352, 11.21750670714594711602515259812

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