Properties

Label 2-189618-1.1-c1-0-39
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 6-s − 2·7-s + 8-s + 9-s − 11-s − 12-s − 2·14-s + 16-s − 17-s + 18-s + 5·19-s + 2·21-s − 22-s − 9·23-s − 24-s − 5·25-s − 27-s − 2·28-s − 5·29-s − 8·31-s + 32-s + 33-s − 34-s + 36-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 1.14·19-s + 0.436·21-s − 0.213·22-s − 1.87·23-s − 0.204·24-s − 25-s − 0.192·27-s − 0.377·28-s − 0.928·29-s − 1.43·31-s + 0.176·32-s + 0.174·33-s − 0.171·34-s + 1/6·36-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: \(2\)
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 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 17 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−13.63737131315883, −13.10919815610885, −12.57621809458468, −12.09624740797219, −11.97383475511316, −11.32447775429867, −10.70541940004746, −10.51833374694544, −9.821507139385344, −9.421985968017313, −9.041313766682820, −8.133139733139794, −7.643025811881015, −7.349882285862567, −6.702109696263543, −6.198503372660628, −5.707472221379045, −5.439752341636529, −4.839934642501249, −4.088774210518142, −3.693992011791533, −3.306143977561409, −2.462851285139705, −1.892925450724718, −1.328514383550533, 0, 0, 1.328514383550533, 1.892925450724718, 2.462851285139705, 3.306143977561409, 3.693992011791533, 4.088774210518142, 4.839934642501249, 5.439752341636529, 5.707472221379045, 6.198503372660628, 6.702109696263543, 7.349882285862567, 7.643025811881015, 8.133139733139794, 9.041313766682820, 9.421985968017313, 9.821507139385344, 10.51833374694544, 10.70541940004746, 11.32447775429867, 11.97383475511316, 12.09624740797219, 12.57621809458468, 13.10919815610885, 13.63737131315883

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