Properties

Label 2-189618-1.1-c1-0-11
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 − 2·5-s − 6-s + 2·7-s − 8-s + 9-s + 2·10-s + 11-s + 12-s − 2·14-s − 2·15-s + 16-s + 17-s − 18-s − 2·19-s − 2·20-s + 2·21-s − 22-s + 2·23-s − 24-s − 25-s + 27-s + 2·28-s + 2·29-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 + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s + 0.288·12-s − 0.534·14-s − 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.458·19-s − 0.447·20-s + 0.436·21-s − 0.213·22-s + 0.417·23-s − 0.204·24-s − 1/5·25-s + 0.192·27-s + 0.377·28-s + 0.371·29-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: $\chi_{189618} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 189618,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.917061036\)
\(L(\frac12)\) \(\approx\) \(1.917061036\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−13.05162559669486, −12.49511933250294, −12.12411404019283, −11.58475129336510, −11.23967425553959, −10.83314989920573, −10.12527344179117, −9.873274147432277, −9.235936906189586, −8.618894439271303, −8.394969992422150, −7.963843038992092, −7.554678959164396, −6.930206125278582, −6.607956580078312, −5.931469947385489, −5.086402057337385, −4.803336436532301, −4.045376007088847, −3.597415066602565, −3.094122635817112, −2.312330158024433, −1.826883905007866, −1.140006403990610, −0.4607082954156123, 0.4607082954156123, 1.140006403990610, 1.826883905007866, 2.312330158024433, 3.094122635817112, 3.597415066602565, 4.045376007088847, 4.803336436532301, 5.086402057337385, 5.931469947385489, 6.607956580078312, 6.930206125278582, 7.554678959164396, 7.963843038992092, 8.394969992422150, 8.618894439271303, 9.235936906189586, 9.873274147432277, 10.12527344179117, 10.83314989920573, 11.23967425553959, 11.58475129336510, 12.12411404019283, 12.49511933250294, 13.05162559669486

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