Properties

Label 2-198-1.1-c1-0-4
Degree $2$
Conductor $198$
Sign $-1$
Analytic cond. $1.58103$
Root an. cond. $1.25739$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 2·5-s − 4·7-s − 8-s + 2·10-s + 11-s − 6·13-s + 4·14-s + 16-s − 2·17-s + 4·19-s − 2·20-s − 22-s − 4·23-s − 25-s + 6·26-s − 4·28-s − 6·29-s − 32-s + 2·34-s + 8·35-s + 6·37-s − 4·38-s + 2·40-s + 6·41-s + 4·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s − 1.51·7-s − 0.353·8-s + 0.632·10-s + 0.301·11-s − 1.66·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.447·20-s − 0.213·22-s − 0.834·23-s − 1/5·25-s + 1.17·26-s − 0.755·28-s − 1.11·29-s − 0.176·32-s + 0.342·34-s + 1.35·35-s + 0.986·37-s − 0.648·38-s + 0.316·40-s + 0.937·41-s + 0.609·43-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(1.58103\)
Root analytic conductor: \(1.25739\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 198,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
11 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−12.06831548180719701984563530619, −10.95942742879596868640115475077, −9.695426551233672906693226319091, −9.331429543089205578507139477048, −7.76149770434504125566929280206, −7.14262372141687320205238561411, −5.90428105253920776614679871565, −4.09304077381815838984743047096, −2.74090516225731871253746662847, 0, 2.74090516225731871253746662847, 4.09304077381815838984743047096, 5.90428105253920776614679871565, 7.14262372141687320205238561411, 7.76149770434504125566929280206, 9.331429543089205578507139477048, 9.695426551233672906693226319091, 10.95942742879596868640115475077, 12.06831548180719701984563530619

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