Properties

Label 2-189618-1.1-c1-0-29
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 − 2·5-s + 6-s − 7-s + 8-s + 9-s − 2·10-s − 11-s + 12-s − 14-s − 2·15-s + 16-s − 17-s + 18-s + 4·19-s − 2·20-s − 21-s − 22-s − 6·23-s + 24-s − 25-s + 27-s − 28-s − 6·29-s − 2·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s + 0.288·12-s − 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.218·21-s − 0.213·22-s − 1.25·23-s + 0.204·24-s − 1/5·25-s + 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.365·30-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: Trivial
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 \( 1 + 2 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
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

−13.35067275790480, −12.86295669317117, −12.53373852879626, −12.00345189669233, −11.50700024721743, −11.14572900364113, −10.69152871842050, −9.926270480773207, −9.583603184871406, −9.215294828892630, −8.352872253549553, −7.973710156635226, −7.684800499371684, −7.119859765965690, −6.638483886451748, −6.024478334448759, −5.445975940805274, −5.042225678995937, −4.212856716044062, −3.864991918671456, −3.557254352893917, −2.874077115315608, −2.289626055201310, −1.755594830835895, −0.8103544642618587, 0, 0.8103544642618587, 1.755594830835895, 2.289626055201310, 2.874077115315608, 3.557254352893917, 3.864991918671456, 4.212856716044062, 5.042225678995937, 5.445975940805274, 6.024478334448759, 6.638483886451748, 7.119859765965690, 7.684800499371684, 7.973710156635226, 8.352872253549553, 9.215294828892630, 9.583603184871406, 9.926270480773207, 10.69152871842050, 11.14572900364113, 11.50700024721743, 12.00345189669233, 12.53373852879626, 12.86295669317117, 13.35067275790480

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