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Properties

Label 2-198-11.3-c1-0-4
Degree $2$
Conductor $198$
Sign $-0.116 + 0.993i$
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−3.11 − 2.26i)5-s + (0.736 − 2.26i)7-s + (−0.309 − 0.951i)8-s − 3.85·10-s + (3.23 − 0.726i)11-s + (−1 + 0.726i)13-s + (−0.736 − 2.26i)14-s + (−0.809 − 0.587i)16-s + (5.23 + 3.80i)17-s + (0.381 + 1.17i)19-s + (−3.11 + 2.26i)20-s + (2.19 − 2.48i)22-s − 1.23·23-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (−1.39 − 1.01i)5-s + (0.278 − 0.856i)7-s + (−0.109 − 0.336i)8-s − 1.21·10-s + (0.975 − 0.219i)11-s + (−0.277 + 0.201i)13-s + (−0.196 − 0.605i)14-s + (−0.202 − 0.146i)16-s + (1.26 + 0.922i)17-s + (0.0876 + 0.269i)19-s + (−0.697 + 0.506i)20-s + (0.467 − 0.530i)22-s − 0.257·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $-0.116 + 0.993i$
Analytic conductor: \(1.58103\)
Root analytic conductor: \(1.25739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :1/2),\ -0.116 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.889927 - 1.00073i\)
\(L(\frac12)\) \(\approx\) \(0.889927 - 1.00073i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 \)
11 \( 1 + (-3.23 + 0.726i)T \)
good5 \( 1 + (3.11 + 2.26i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.736 + 2.26i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1 - 0.726i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-5.23 - 3.80i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.381 - 1.17i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.23T + 23T^{2} \)
29 \( 1 + (-0.809 + 2.48i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.16 - 3.75i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.23 + 9.95i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.381 + 1.17i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.23T + 43T^{2} \)
47 \( 1 + (-2 - 6.15i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.30 + 1.67i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.881 - 2.71i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-9.09 - 6.60i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + (-2.85 - 2.07i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.33 + 4.11i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.11 - 3.71i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (13.2 + 9.59i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 1.70T + 89T^{2} \)
97 \( 1 + (-3.5 + 2.54i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.22633674648987573587004372548, −11.50731742532321626619233793260, −10.53775666741804285871893084490, −9.235859429159395783082511272450, −8.104398258895650137927878650208, −7.21912135647023295774099480307, −5.59990951559869982322633200425, −4.21755485330225947931972780307, −3.75684232007856290983698857449, −1.14018290067349941703985030644, 2.87873170426480198592272856752, 3.95445723247350060552542679705, 5.30875235929324402767889025313, 6.69037171036422832220222722274, 7.49714429107963219216083118002, 8.421658230730484425093525056349, 9.789488535747253379370345228840, 11.27858307603748748503666775767, 11.78958540053274919160008721658, 12.47072194912552914752849513409

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