Properties

Label 2-189618-1.1-c1-0-40
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 − 2·5-s − 6-s + 8-s + 9-s − 2·10-s + 11-s − 12-s + 2·15-s + 16-s + 17-s + 18-s − 6·19-s − 2·20-s + 22-s + 4·23-s − 24-s − 25-s − 27-s − 6·29-s + 2·30-s − 8·31-s + 32-s − 33-s + 34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s − 0.288·12-s + 0.516·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 1.37·19-s − 0.447·20-s + 0.213·22-s + 0.834·23-s − 0.204·24-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.365·30-s − 1.43·31-s + 0.176·32-s − 0.174·33-s + 0.171·34-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 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.40421226062398, −12.98224826413446, −12.66704057743945, −12.22719458028211, −11.65639260530501, −11.34777368261771, −10.92395685851953, −10.54093219584535, −9.926628025886606, −9.320343982339784, −8.800417201743020, −8.253594283398501, −7.712883464035486, −7.238993899380646, −6.778095388674030, −6.341304820013476, −5.746913586561580, −5.169908912938974, −4.838416437254241, −4.100415200571676, −3.761379962733287, −3.342719041001750, −2.536926521718325, −1.756704036443752, −1.346599479650423, 0, 0, 1.346599479650423, 1.756704036443752, 2.536926521718325, 3.342719041001750, 3.761379962733287, 4.100415200571676, 4.838416437254241, 5.169908912938974, 5.746913586561580, 6.341304820013476, 6.778095388674030, 7.238993899380646, 7.712883464035486, 8.253594283398501, 8.800417201743020, 9.320343982339784, 9.926628025886606, 10.54093219584535, 10.92395685851953, 11.34777368261771, 11.65639260530501, 12.22719458028211, 12.66704057743945, 12.98224826413446, 13.40421226062398

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