Properties

Label 2-189618-1.1-c1-0-44
Degree $2$
Conductor $189618$
Sign $-1$
Analytic cond. $1514.10$
Root an. cond. $38.9115$
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 + 3-s + 4-s + 4·5-s + 6-s + 2·7-s + 8-s + 9-s + 4·10-s − 11-s + 12-s + 2·14-s + 4·15-s + 16-s − 17-s + 18-s + 4·19-s + 4·20-s + 2·21-s − 22-s − 6·23-s + 24-s + 11·25-s + 27-s + 2·28-s + 4·30-s − 10·31-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.301·11-s + 0.288·12-s + 0.534·14-s + 1.03·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.894·20-s + 0.436·21-s − 0.213·22-s − 1.25·23-s + 0.204·24-s + 11/5·25-s + 0.192·27-s + 0.377·28-s + 0.730·30-s − 1.79·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189618\)    =    \(2 \cdot 3 \cdot 11 \cdot 13^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(1514.10\)
Root analytic conductor: \(38.9115\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{189618} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 189618,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 \)
17 \( 1 + T \)
good5 \( 1 - 4 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.39730895900392, −13.19217556593258, −12.42646303114176, −12.14159723197948, −11.54519527103370, −10.88635925937081, −10.47631611909927, −10.13482540456920, −9.548232905397329, −9.186601354482814, −8.574471383184925, −8.175710748974199, −7.457674902018001, −7.051536509370709, −6.536806865130529, −5.887347591457504, −5.464386661813196, −5.147950293083800, −4.620398728322291, −3.860942797040134, −3.296454145318397, −2.762693242867377, −2.051661825818879, −1.702689042770721, −1.390008379515225, 0, 1.390008379515225, 1.702689042770721, 2.051661825818879, 2.762693242867377, 3.296454145318397, 3.860942797040134, 4.620398728322291, 5.147950293083800, 5.464386661813196, 5.887347591457504, 6.536806865130529, 7.051536509370709, 7.457674902018001, 8.175710748974199, 8.574471383184925, 9.186601354482814, 9.548232905397329, 10.13482540456920, 10.47631611909927, 10.88635925937081, 11.54519527103370, 12.14159723197948, 12.42646303114176, 13.19217556593258, 13.39730895900392

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