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Properties

Label 2-198-11.5-c1-0-3
Degree $2$
Conductor $198$
Sign $0.0219 + 0.999i$
Analytic cond. $1.58103$
Root an. cond. $1.25739$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.381 − 1.17i)5-s + (1.61 − 1.17i)7-s + (0.809 + 0.587i)8-s − 1.23·10-s + (0.809 − 3.21i)11-s + (−1 − 3.07i)13-s + (−1.61 − 1.17i)14-s + (0.309 − 0.951i)16-s + (−0.5 + 1.53i)17-s + (−0.690 − 0.502i)19-s + (0.381 + 1.17i)20-s + (−3.30 + 0.224i)22-s + 3.23·23-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.404 + 0.293i)4-s + (0.170 − 0.525i)5-s + (0.611 − 0.444i)7-s + (0.286 + 0.207i)8-s − 0.390·10-s + (0.243 − 0.969i)11-s + (−0.277 − 0.853i)13-s + (−0.432 − 0.314i)14-s + (0.0772 − 0.237i)16-s + (−0.121 + 0.373i)17-s + (−0.158 − 0.115i)19-s + (0.0854 + 0.262i)20-s + (−0.705 + 0.0478i)22-s + 0.674·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $0.0219 + 0.999i$
Analytic conductor: \(1.58103\)
Root analytic conductor: \(1.25739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :1/2),\ 0.0219 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.778448 - 0.761519i\)
\(L(\frac12)\) \(\approx\) \(0.778448 - 0.761519i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
3 \( 1 \)
11 \( 1 + (-0.809 + 3.21i)T \)
good5 \( 1 + (-0.381 + 1.17i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.61 + 1.17i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1 + 3.07i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.5 - 1.53i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.690 + 0.502i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 3.23T + 23T^{2} \)
29 \( 1 + (3.61 - 2.62i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.618 - 1.90i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (7.85 - 5.70i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-2.73 - 1.98i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 + (-2 - 1.45i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.23 - 9.95i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (5.16 - 3.75i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2 + 6.15i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 0.0901T + 67T^{2} \)
71 \( 1 + (-0.236 + 0.726i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (10.2 - 7.41i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (4.14 + 12.7i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.95 + 6.01i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 3.09T + 89T^{2} \)
97 \( 1 + (-4.28 - 13.1i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.25319014022772692321276419172, −11.05255102791470322956082864495, −10.56244262175845292374075063984, −9.198384613110928096824399470594, −8.463405069865109698003870236452, −7.36293779121668347779159799637, −5.69962206984283101155872303836, −4.55180876232166857369471387970, −3.11639129397682993269913876966, −1.19739128417084498318473961457, 2.16964224035535640948756652804, 4.26347611165426583986142383383, 5.41786008203342927964316730388, 6.73225149992110496264276854049, 7.45196193568850040579854831301, 8.764388546926799255850885258379, 9.563777288649455714984714966276, 10.68521318114929436201995818362, 11.72563183840060063499681636811, 12.73209737577653920141385395408

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