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Properties

Label 2-198-33.2-c1-0-1
Degree $2$
Conductor $198$
Sign $0.949 - 0.313i$
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 + (1.05 − 0.343i)5-s + (2.97 − 4.09i)7-s + (0.809 − 0.587i)8-s + 1.11i·10-s + (0.0444 + 3.31i)11-s + (−0.0598 − 0.0194i)13-s + (2.97 + 4.09i)14-s + (0.309 + 0.951i)16-s + (0.486 + 1.49i)17-s + (2.74 + 3.78i)19-s + (−1.05 − 0.343i)20-s + (−3.16 − 0.982i)22-s − 7.56i·23-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.404 − 0.293i)4-s + (0.473 − 0.153i)5-s + (1.12 − 1.54i)7-s + (0.286 − 0.207i)8-s + 0.351i·10-s + (0.0134 + 0.999i)11-s + (−0.0165 − 0.00539i)13-s + (0.795 + 1.09i)14-s + (0.0772 + 0.237i)16-s + (0.117 + 0.362i)17-s + (0.630 + 0.868i)19-s + (−0.236 − 0.0768i)20-s + (−0.675 − 0.209i)22-s − 1.57i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $0.949 - 0.313i$
Analytic conductor: \(1.58103\)
Root analytic conductor: \(1.25739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :1/2),\ 0.949 - 0.313i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18875 + 0.190878i\)
\(L(\frac12)\) \(\approx\) \(1.18875 + 0.190878i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0444 - 3.31i)T \)
good5 \( 1 + (-1.05 + 0.343i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-2.97 + 4.09i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.0598 + 0.0194i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.486 - 1.49i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.74 - 3.78i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 7.56iT - 23T^{2} \)
29 \( 1 + (2.03 + 1.47i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.789 + 2.42i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.513 - 0.373i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (6.87 - 4.99i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.52iT - 43T^{2} \)
47 \( 1 + (0.490 + 0.675i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (5.23 + 1.69i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.96 + 8.20i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (12.2 - 3.97i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 2.91T + 67T^{2} \)
71 \( 1 + (10.5 - 3.43i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.19 - 5.77i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-10.7 - 3.49i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.48 - 10.7i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 11.4iT - 89T^{2} \)
97 \( 1 + (-1.30 + 4.03i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.75370687754624975315875091426, −11.40781416785749188453992318659, −10.31555284248187016970718119839, −9.706835044104230989201056642253, −8.187209147398608167087076554892, −7.55901776252870710055043408717, −6.46384526584611678241961659069, −5.02776702547045387464350928416, −4.12295751726920488955379584983, −1.52632511305857491550958194953, 1.85241496754441714594993467720, 3.14777750805293213732827916043, 5.05796908320357559355962525470, 5.83327442282277845639111453198, 7.61739704582805904545265474988, 8.745536571626525062109855079755, 9.304835619819717838506665554376, 10.63689345880464967492806870169, 11.62920485421343036406821658959, 11.98530834040668240343581128093

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