Properties

Label 2-18-1.1-c11-0-2
Degree $2$
Conductor $18$
Sign $1$
Analytic cond. $13.8301$
Root an. cond. $3.71889$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 32·2-s + 1.02e3·4-s + 1.17e4·5-s − 5.00e4·7-s + 3.27e4·8-s + 3.75e5·10-s + 5.31e5·11-s + 1.33e6·13-s − 1.60e6·14-s + 1.04e6·16-s + 5.10e6·17-s + 2.90e6·19-s + 1.20e7·20-s + 1.70e7·22-s − 3.05e7·23-s + 8.87e7·25-s + 4.26e7·26-s − 5.12e7·28-s + 7.70e7·29-s − 2.39e8·31-s + 3.35e7·32-s + 1.63e8·34-s − 5.86e8·35-s − 7.85e8·37-s + 9.28e7·38-s + 3.84e8·40-s − 4.11e8·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.67·5-s − 1.12·7-s + 0.353·8-s + 1.18·10-s + 0.994·11-s + 0.995·13-s − 0.795·14-s + 1/4·16-s + 0.872·17-s + 0.268·19-s + 0.839·20-s + 0.703·22-s − 0.991·23-s + 1.81·25-s + 0.703·26-s − 0.562·28-s + 0.697·29-s − 1.50·31-s + 0.176·32-s + 0.617·34-s − 1.88·35-s − 1.86·37-s + 0.190·38-s + 0.593·40-s − 0.554·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18\)    =    \(2 \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(13.8301\)
Root analytic conductor: \(3.71889\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 18,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(3.481310618\)
\(L(\frac12)\) \(\approx\) \(3.481310618\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{5} T \)
3 \( 1 \)
good5 \( 1 - 2346 p T + p^{11} T^{2} \)
7 \( 1 + 7144 p T + p^{11} T^{2} \)
11 \( 1 - 531420 T + p^{11} T^{2} \)
13 \( 1 - 1332566 T + p^{11} T^{2} \)
17 \( 1 - 5109678 T + p^{11} T^{2} \)
19 \( 1 - 2901404 T + p^{11} T^{2} \)
23 \( 1 + 30597000 T + p^{11} T^{2} \)
29 \( 1 - 77006634 T + p^{11} T^{2} \)
31 \( 1 + 239418352 T + p^{11} T^{2} \)
37 \( 1 + 785041666 T + p^{11} T^{2} \)
41 \( 1 + 411252954 T + p^{11} T^{2} \)
43 \( 1 - 351233348 T + p^{11} T^{2} \)
47 \( 1 + 95821680 T + p^{11} T^{2} \)
53 \( 1 - 1465857378 T + p^{11} T^{2} \)
59 \( 1 + 5621152020 T + p^{11} T^{2} \)
61 \( 1 + 10473587770 T + p^{11} T^{2} \)
67 \( 1 - 4515307532 T + p^{11} T^{2} \)
71 \( 1 - 8509579560 T + p^{11} T^{2} \)
73 \( 1 - 2012496986 T + p^{11} T^{2} \)
79 \( 1 + 22238409568 T + p^{11} T^{2} \)
83 \( 1 + 6328647516 T + p^{11} T^{2} \)
89 \( 1 - 50123706678 T + p^{11} T^{2} \)
97 \( 1 - 94805961314 T + p^{11} T^{2} \)
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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

−16.05450114493974296068423188073, −14.26770594204554709843088145895, −13.51628284255213760889327899401, −12.31015133449301585407918873312, −10.36840032055032392221638585038, −9.224191660655688862679435719552, −6.57959580553579358598933389693, −5.68971361281167691539568307649, −3.41771560822878169631254205535, −1.60457919752433608316953414142, 1.60457919752433608316953414142, 3.41771560822878169631254205535, 5.68971361281167691539568307649, 6.57959580553579358598933389693, 9.224191660655688862679435719552, 10.36840032055032392221638585038, 12.31015133449301585407918873312, 13.51628284255213760889327899401, 14.26770594204554709843088145895, 16.05450114493974296068423188073

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