Properties

Label 2-189618-1.1-c1-0-7
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 3·7-s − 8-s + 9-s − 11-s − 12-s + 3·14-s + 16-s − 17-s − 18-s − 6·19-s + 3·21-s + 22-s + 24-s − 5·25-s − 27-s − 3·28-s + 6·29-s − 32-s + 33-s + 34-s + 36-s + 3·37-s + 6·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 0.801·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.37·19-s + 0.654·21-s + 0.213·22-s + 0.204·24-s − 25-s − 0.192·27-s − 0.566·28-s + 1.11·29-s − 0.176·32-s + 0.174·33-s + 0.171·34-s + 1/6·36-s + 0.493·37-s + 0.973·38-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: \(0\)
Selberg data: \((2,\ 189618,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8486414983\)
\(L(\frac12)\) \(\approx\) \(0.8486414983\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−12.91707510960277, −12.65940122009894, −11.99597021620348, −11.87877910495464, −11.01373508685144, −10.57586219213828, −10.50402337554713, −9.675637003808099, −9.492144115934789, −8.919132569685527, −8.336394599132650, −7.917617252050303, −7.269940853080808, −6.814800075219978, −6.402487733057667, −5.859514856232382, −5.614764648306863, −4.681659930191286, −4.148549003688299, −3.736147400077463, −2.805005116893289, −2.484400680790480, −1.819196930663966, −0.8534091206802280, −0.3941937048398590, 0.3941937048398590, 0.8534091206802280, 1.819196930663966, 2.484400680790480, 2.805005116893289, 3.736147400077463, 4.148549003688299, 4.681659930191286, 5.614764648306863, 5.859514856232382, 6.402487733057667, 6.814800075219978, 7.269940853080808, 7.917617252050303, 8.336394599132650, 8.919132569685527, 9.492144115934789, 9.675637003808099, 10.50402337554713, 10.57586219213828, 11.01373508685144, 11.87877910495464, 11.99597021620348, 12.65940122009894, 12.91707510960277

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