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Properties

Label 2-198-11.4-c5-0-9
Degree $2$
Conductor $198$
Sign $-0.505 - 0.863i$
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  + (3.23 + 2.35i)2-s + (4.94 + 15.2i)4-s + (34.0 − 24.7i)5-s + (58.4 + 179. i)7-s + (−19.7 + 60.8i)8-s + 168.·10-s + (−393. + 76.3i)11-s + (193. + 140. i)13-s + (−233. + 719. i)14-s + (−207. + 150. i)16-s + (−9.75 + 7.08i)17-s + (1.49 − 4.59i)19-s + (544. + 395. i)20-s + (−1.45e3 − 679. i)22-s + 898.·23-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.154 + 0.475i)4-s + (0.608 − 0.442i)5-s + (0.450 + 1.38i)7-s + (−0.109 + 0.336i)8-s + 0.531·10-s + (−0.981 + 0.190i)11-s + (0.316 + 0.230i)13-s + (−0.318 + 0.980i)14-s + (−0.202 + 0.146i)16-s + (−0.00818 + 0.00594i)17-s + (0.000949 − 0.00292i)19-s + (0.304 + 0.221i)20-s + (−0.640 − 0.299i)22-s + 0.354·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 - 0.863i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.505 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $-0.505 - 0.863i$
Analytic conductor: \(31.7559\)
Root analytic conductor: \(5.63524\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :5/2),\ -0.505 - 0.863i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.671867755\)
\(L(\frac12)\) \(\approx\) \(2.671867755\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.23 - 2.35i)T \)
3 \( 1 \)
11 \( 1 + (393. - 76.3i)T \)
good5 \( 1 + (-34.0 + 24.7i)T + (965. - 2.97e3i)T^{2} \)
7 \( 1 + (-58.4 - 179. i)T + (-1.35e4 + 9.87e3i)T^{2} \)
13 \( 1 + (-193. - 140. i)T + (1.14e5 + 3.53e5i)T^{2} \)
17 \( 1 + (9.75 - 7.08i)T + (4.38e5 - 1.35e6i)T^{2} \)
19 \( 1 + (-1.49 + 4.59i)T + (-2.00e6 - 1.45e6i)T^{2} \)
23 \( 1 - 898.T + 6.43e6T^{2} \)
29 \( 1 + (-623. - 1.91e3i)T + (-1.65e7 + 1.20e7i)T^{2} \)
31 \( 1 + (95.7 + 69.5i)T + (8.84e6 + 2.72e7i)T^{2} \)
37 \( 1 + (-3.97e3 - 1.22e4i)T + (-5.61e7 + 4.07e7i)T^{2} \)
41 \( 1 + (3.91e3 - 1.20e4i)T + (-9.37e7 - 6.80e7i)T^{2} \)
43 \( 1 + 1.40e4T + 1.47e8T^{2} \)
47 \( 1 + (-2.12e3 + 6.54e3i)T + (-1.85e8 - 1.34e8i)T^{2} \)
53 \( 1 + (5.95e3 + 4.32e3i)T + (1.29e8 + 3.97e8i)T^{2} \)
59 \( 1 + (2.33e3 + 7.19e3i)T + (-5.78e8 + 4.20e8i)T^{2} \)
61 \( 1 + (1.96e4 - 1.42e4i)T + (2.60e8 - 8.03e8i)T^{2} \)
67 \( 1 + 1.12e4T + 1.35e9T^{2} \)
71 \( 1 + (-5.36e4 + 3.89e4i)T + (5.57e8 - 1.71e9i)T^{2} \)
73 \( 1 + (-2.74e4 - 8.43e4i)T + (-1.67e9 + 1.21e9i)T^{2} \)
79 \( 1 + (-1.15e4 - 8.36e3i)T + (9.50e8 + 2.92e9i)T^{2} \)
83 \( 1 + (-9.62e4 + 6.99e4i)T + (1.21e9 - 3.74e9i)T^{2} \)
89 \( 1 + 2.77e4T + 5.58e9T^{2} \)
97 \( 1 + (8.79e4 + 6.38e4i)T + (2.65e9 + 8.16e9i)T^{2} \)
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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

−12.09691305554728592483065841502, −11.19803630682194535527730186167, −9.811193613124636294530610670562, −8.764628087728467303531825187464, −7.976036041858473875823368661273, −6.50234714506122101846366493861, −5.45064565552126165562453014310, −4.84808237506246571490888575035, −2.98300778915398979550453282248, −1.77172516797547203022949453943, 0.64904600301461042964999284770, 2.14935106015318891672659812578, 3.50120292460975341427723674003, 4.70951373542822122229255109959, 5.87477717365584973336221490698, 7.05387190539099731769472863295, 8.078204115986168166590374737615, 9.659108520283950009301092898709, 10.66471243127921677257626702752, 10.91008050920807973286914444154

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