Properties

Label 2-189618-1.1-c1-0-15
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 − 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 + 19-s − 4·20-s + 2·21-s − 22-s + 7·23-s + 24-s + 11·25-s + 27-s + 2·28-s − 29-s − 4·30-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 + 0.229·19-s − 0.894·20-s + 0.436·21-s − 0.213·22-s + 1.45·23-s + 0.204·24-s + 11/5·25-s + 0.192·27-s + 0.377·28-s − 0.185·29-s − 0.730·30-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: \(0\)
Selberg data: \((2,\ 189618,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.516150288\)
\(L(\frac12)\) \(\approx\) \(4.516150288\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−13.10401770427002, −12.62248660192897, −12.13984274561870, −11.76364664391964, −11.37451747505108, −10.89859010320566, −10.49956437406420, −9.923220044791918, −9.065836113053528, −8.767062942996767, −8.128576257512179, −7.851161277234023, −7.452208861396177, −6.932055443916562, −6.496580837650973, −5.649723917436906, −4.953279502320924, −4.702299091403668, −4.280624415903143, −3.521107596840553, −3.250550269133491, −2.749476841014114, −1.949407704104954, −1.171783225545526, −0.5606047909444037, 0.5606047909444037, 1.171783225545526, 1.949407704104954, 2.749476841014114, 3.250550269133491, 3.521107596840553, 4.280624415903143, 4.702299091403668, 4.953279502320924, 5.649723917436906, 6.496580837650973, 6.932055443916562, 7.452208861396177, 7.851161277234023, 8.128576257512179, 8.767062942996767, 9.065836113053528, 9.923220044791918, 10.49956437406420, 10.89859010320566, 11.37451747505108, 11.76364664391964, 12.13984274561870, 12.62248660192897, 13.10401770427002

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