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Properties

Label 2-198-9.4-c5-0-15
Degree $2$
Conductor $198$
Sign $-0.996 + 0.0876i$
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 + (15.2 − 3.37i)3-s + (−7.99 + 13.8i)4-s + (−52.1 + 90.3i)5-s + (42.1 + 45.9i)6-s + (64.3 + 111. i)7-s − 63.9·8-s + (220. − 102. i)9-s − 417.·10-s + (−60.5 − 104. i)11-s + (−74.9 + 237. i)12-s + (−230. + 399. i)13-s + (−257. + 445. i)14-s + (−488. + 1.55e3i)15-s + (−128 − 221. i)16-s − 63.6·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.976 − 0.216i)3-s + (−0.249 + 0.433i)4-s + (−0.933 + 1.61i)5-s + (0.477 + 0.521i)6-s + (0.496 + 0.859i)7-s − 0.353·8-s + (0.906 − 0.423i)9-s − 1.32·10-s + (−0.150 − 0.261i)11-s + (−0.150 + 0.476i)12-s + (−0.378 + 0.656i)13-s + (−0.350 + 0.607i)14-s + (−0.561 + 1.78i)15-s + (−0.125 − 0.216i)16-s − 0.0534·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0876i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $-0.996 + 0.0876i$
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: \((2,\ 198,\ (\ :5/2),\ -0.996 + 0.0876i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.160004264\)
\(L(\frac12)\) \(\approx\) \(2.160004264\)
\(L(\frac{7}{2})\) 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 \( 1 + (-15.2 + 3.37i)T \)
11 \( 1 + (60.5 + 104. i)T \)
good5 \( 1 + (52.1 - 90.3i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-64.3 - 111. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
13 \( 1 + (230. - 399. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + 63.6T + 1.41e6T^{2} \)
19 \( 1 + 110.T + 2.47e6T^{2} \)
23 \( 1 + (321. - 556. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (1.85e3 + 3.21e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (786. - 1.36e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 6.91e3T + 6.93e7T^{2} \)
41 \( 1 + (8.72e3 - 1.51e4i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (7.32e3 + 1.26e4i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-1.03e4 - 1.78e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + 2.31e4T + 4.18e8T^{2} \)
59 \( 1 + (-1.60e3 + 2.78e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-2.58e4 - 4.47e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.40e4 - 4.16e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 6.20e4T + 1.80e9T^{2} \)
73 \( 1 - 3.69e4T + 2.07e9T^{2} \)
79 \( 1 + (1.43e4 + 2.49e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (3.59e3 + 6.22e3i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 - 5.62e4T + 5.58e9T^{2} \)
97 \( 1 + (-2.30e4 - 3.98e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−12.02539426032974786466029747428, −11.37842984344395461880886533441, −10.03520332301531493143510981354, −8.768480960040047188593288301569, −7.85278369876222080102577970010, −7.16278521965404806844990343279, −6.15310820619497696547157051416, −4.38567543196213644264176489969, −3.26251281215237710850271113233, −2.30870289131199874406493691051, 0.50883315275035854429836158132, 1.72250731360519324854433647056, 3.49169986945964811544503792260, 4.41093352910252236591553037650, 5.10481059248837013414050795475, 7.42851896989875760620920087832, 8.170511892425522741092815661440, 9.054114164173774709046950795202, 10.07414396226450200480883709329, 11.14899085213860547044179075556

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