Properties

Label 2-198-1.1-c9-0-30
Degree $2$
Conductor $198$
Sign $-1$
Analytic cond. $101.977$
Root an. cond. $10.0983$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·2-s + 256·4-s − 1.18e3·5-s + 1.17e3·7-s + 4.09e3·8-s − 1.89e4·10-s − 1.46e4·11-s + 8.24e4·13-s + 1.87e4·14-s + 6.55e4·16-s − 4.40e5·17-s + 8.04e5·19-s − 3.03e5·20-s − 2.34e5·22-s + 2.28e5·23-s − 5.47e5·25-s + 1.31e6·26-s + 3.00e5·28-s − 2.28e6·29-s + 2.56e6·31-s + 1.04e6·32-s − 7.05e6·34-s − 1.39e6·35-s − 4.68e6·37-s + 1.28e7·38-s − 4.85e6·40-s − 3.30e6·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.848·5-s + 0.184·7-s + 0.353·8-s − 0.599·10-s − 0.301·11-s + 0.800·13-s + 0.130·14-s + 0.250·16-s − 1.28·17-s + 1.41·19-s − 0.424·20-s − 0.213·22-s + 0.170·23-s − 0.280·25-s + 0.566·26-s + 0.0924·28-s − 0.599·29-s + 0.498·31-s + 0.176·32-s − 0.905·34-s − 0.156·35-s − 0.411·37-s + 1.00·38-s − 0.299·40-s − 0.182·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(101.977\)
Root analytic conductor: \(10.0983\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 198,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 16T \)
3 \( 1 \)
11 \( 1 + 1.46e4T \)
good5 \( 1 + 1.18e3T + 1.95e6T^{2} \)
7 \( 1 - 1.17e3T + 4.03e7T^{2} \)
13 \( 1 - 8.24e4T + 1.06e10T^{2} \)
17 \( 1 + 4.40e5T + 1.18e11T^{2} \)
19 \( 1 - 8.04e5T + 3.22e11T^{2} \)
23 \( 1 - 2.28e5T + 1.80e12T^{2} \)
29 \( 1 + 2.28e6T + 1.45e13T^{2} \)
31 \( 1 - 2.56e6T + 2.64e13T^{2} \)
37 \( 1 + 4.68e6T + 1.29e14T^{2} \)
41 \( 1 + 3.30e6T + 3.27e14T^{2} \)
43 \( 1 - 7.81e6T + 5.02e14T^{2} \)
47 \( 1 - 2.85e6T + 1.11e15T^{2} \)
53 \( 1 + 9.97e6T + 3.29e15T^{2} \)
59 \( 1 + 6.31e7T + 8.66e15T^{2} \)
61 \( 1 + 2.12e8T + 1.16e16T^{2} \)
67 \( 1 + 2.27e8T + 2.72e16T^{2} \)
71 \( 1 - 1.55e8T + 4.58e16T^{2} \)
73 \( 1 + 1.04e8T + 5.88e16T^{2} \)
79 \( 1 + 2.96e8T + 1.19e17T^{2} \)
83 \( 1 - 7.38e8T + 1.86e17T^{2} \)
89 \( 1 + 9.00e8T + 3.50e17T^{2} \)
97 \( 1 - 1.29e9T + 7.60e17T^{2} \)
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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

−10.71858799463182567963262821577, −9.292219758533295041001609635090, −8.119676511713394713550796783069, −7.26301568925839135192570903278, −6.11289864116375646139058154253, −4.92625497950434354924992429821, −3.94725522397425187339296481114, −2.94190004612741043061326678379, −1.47701974900509870972813318162, 0, 1.47701974900509870972813318162, 2.94190004612741043061326678379, 3.94725522397425187339296481114, 4.92625497950434354924992429821, 6.11289864116375646139058154253, 7.26301568925839135192570903278, 8.119676511713394713550796783069, 9.292219758533295041001609635090, 10.71858799463182567963262821577

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