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Properties

Label 2-198-9.7-c5-0-15
Degree $2$
Conductor $198$
Sign $0.996 + 0.0798i$
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 + (3.31 − 15.2i)3-s + (−7.99 − 13.8i)4-s + (25.8 + 44.7i)5-s + (−46.1 − 41.9i)6-s + (−53.8 + 93.2i)7-s − 63.9·8-s + (−220. − 101. i)9-s + 206.·10-s + (−60.5 + 104. i)11-s + (−237. + 75.8i)12-s + (315. + 545. i)13-s + (215. + 373. i)14-s + (767. − 245. i)15-s + (−128 + 221. i)16-s + 1.06e3·17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.212 − 0.977i)3-s + (−0.249 − 0.433i)4-s + (0.462 + 0.801i)5-s + (−0.523 − 0.475i)6-s + (−0.415 + 0.719i)7-s − 0.353·8-s + (−0.909 − 0.416i)9-s + 0.654·10-s + (−0.150 + 0.261i)11-s + (−0.476 + 0.152i)12-s + (0.517 + 0.895i)13-s + (0.293 + 0.508i)14-s + (0.881 − 0.281i)15-s + (−0.125 + 0.216i)16-s + 0.897·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0798i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $0.996 + 0.0798i$
Analytic conductor: \(31.7559\)
Root analytic conductor: \(5.63524\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :5/2),\ 0.996 + 0.0798i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.319502231\)
\(L(\frac12)\) \(\approx\) \(2.319502231\)
\(L(\frac{7}{2})\) 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 \( 1 + (-3.31 + 15.2i)T \)
11 \( 1 + (60.5 - 104. i)T \)
good5 \( 1 + (-25.8 - 44.7i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (53.8 - 93.2i)T + (-8.40e3 - 1.45e4i)T^{2} \)
13 \( 1 + (-315. - 545. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 1.06e3T + 1.41e6T^{2} \)
19 \( 1 - 570.T + 2.47e6T^{2} \)
23 \( 1 + (-1.16e3 - 2.01e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (2.52e3 - 4.37e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (-2.55e3 - 4.42e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 - 5.42e3T + 6.93e7T^{2} \)
41 \( 1 + (4.12e3 + 7.15e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (-3.04e3 + 5.27e3i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (2.97e3 - 5.15e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 - 9.34e3T + 4.18e8T^{2} \)
59 \( 1 + (-4.77e3 - 8.27e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-8.74e3 + 1.51e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (5.42e3 + 9.40e3i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 1.41e4T + 1.80e9T^{2} \)
73 \( 1 + 5.13e4T + 2.07e9T^{2} \)
79 \( 1 + (2.63e4 - 4.56e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (5.45e3 - 9.44e3i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 7.47e4T + 5.58e9T^{2} \)
97 \( 1 + (3.41e4 - 5.90e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−11.79245490963031323144477858500, −10.80114628193338164373160108321, −9.622400543531671500209482053679, −8.746705084676817043660864083420, −7.28945645832421099076376944806, −6.35826775723461178150982482273, −5.41693871262188876225593049280, −3.42852705725633243454672107722, −2.49758319163037722644583337312, −1.34428365382705431362510757489, 0.66045573588373672043731108359, 3.03205759965271307500057356202, 4.13976997931118875590084243625, 5.24752562271329979530450502788, 6.06466630458133939523704828171, 7.69053747389695738590422237763, 8.583171398454996933296778735954, 9.622657168565222277933492984194, 10.36649090229422255360549119897, 11.59201382746104725899019516958

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