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Properties

Label 2-198-11.7-c2-0-1
Degree $2$
Conductor $198$
Sign $-0.860 + 0.508i$
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  + (−0.831 + 1.14i)2-s + (−0.618 − 1.90i)4-s + (−1.74 + 1.26i)5-s + (−1.27 + 0.414i)7-s + (2.68 + 0.874i)8-s − 3.05i·10-s + (−10.9 − 0.273i)11-s + (−6.36 + 8.75i)13-s + (0.586 − 1.80i)14-s + (−3.23 + 2.35i)16-s + (−15.8 − 21.8i)17-s + (−9.95 − 3.23i)19-s + (3.49 + 2.53i)20-s + (9.45 − 12.3i)22-s − 14.1·23-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (−0.154 − 0.475i)4-s + (−0.349 + 0.253i)5-s + (−0.182 + 0.0592i)7-s + (0.336 + 0.109i)8-s − 0.305i·10-s + (−0.999 − 0.0248i)11-s + (−0.489 + 0.673i)13-s + (0.0418 − 0.128i)14-s + (−0.202 + 0.146i)16-s + (−0.934 − 1.28i)17-s + (−0.523 − 0.170i)19-s + (0.174 + 0.126i)20-s + (0.429 − 0.561i)22-s − 0.617·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.508i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $-0.860 + 0.508i$
Analytic conductor: \(5.39510\)
Root analytic conductor: \(2.32273\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :1),\ -0.860 + 0.508i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0224984 - 0.0822640i\)
\(L(\frac12)\) \(\approx\) \(0.0224984 - 0.0822640i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.831 - 1.14i)T \)
3 \( 1 \)
11 \( 1 + (10.9 + 0.273i)T \)
good5 \( 1 + (1.74 - 1.26i)T + (7.72 - 23.7i)T^{2} \)
7 \( 1 + (1.27 - 0.414i)T + (39.6 - 28.8i)T^{2} \)
13 \( 1 + (6.36 - 8.75i)T + (-52.2 - 160. i)T^{2} \)
17 \( 1 + (15.8 + 21.8i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (9.95 + 3.23i)T + (292. + 212. i)T^{2} \)
23 \( 1 + 14.1T + 529T^{2} \)
29 \( 1 + (-11.7 + 3.81i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (20.7 + 15.0i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (9.00 + 27.7i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (-10.1 - 3.30i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 + 41.5iT - 1.84e3T^{2} \)
47 \( 1 + (21.6 - 66.4i)T + (-1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (-48.5 - 35.2i)T + (868. + 2.67e3i)T^{2} \)
59 \( 1 + (-16.2 - 49.8i)T + (-2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (-64.7 - 89.1i)T + (-1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 - 60.0T + 4.48e3T^{2} \)
71 \( 1 + (-56.0 + 40.7i)T + (1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (69.7 - 22.6i)T + (4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (58.9 - 81.1i)T + (-1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (59.8 + 82.4i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 - 55.9T + 7.92e3T^{2} \)
97 \( 1 + (-141. - 102. i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−12.94135817372651288907365832189, −11.67698090644944806135097462338, −10.81396235535089496536696199677, −9.708078930955208192115263728895, −8.828332738471788005711081254921, −7.61265847195073226198235174010, −6.91031608547986572849487859195, −5.57596774903246388509429806941, −4.34274325276307722990796546109, −2.47217721292571897493449970407, 0.05203291511395295521482444695, 2.20465576774539150891464820502, 3.72478339363313436002620278025, 5.03887981668602950527777509144, 6.56400685990676342303586384545, 7.999629917276912053528899298589, 8.505858350924790001752493698639, 9.980958667835607625218479201437, 10.54307995021511710217260138591, 11.62853211877084825356454011775

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