Properties

Label 2-189618-1.1-c1-0-34
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 $1$

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 − 6·19-s + 2·20-s + 2·21-s − 22-s − 6·23-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 − 1.37·19-s + 0.447·20-s + 0.436·21-s − 0.213·22-s − 1.25·23-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: \(1\)
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 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−13.28582320534647, −13.04599885275312, −12.63921529798052, −11.97817377303267, −11.49726381707720, −11.01851520965774, −10.57627243468486, −9.995170347266376, −9.713415430654421, −9.046699565852487, −8.506426462748760, −8.114249566612408, −7.601349626115677, −7.099791091246954, −6.420908282495581, −5.987982267221447, −5.604846211664618, −4.988041655057666, −4.351894087882593, −4.063179351509293, −3.439863157095992, −2.521842938308500, −2.253262293058802, −1.846216532620069, −1.115981504687644, 0, 1.115981504687644, 1.846216532620069, 2.253262293058802, 2.521842938308500, 3.439863157095992, 4.063179351509293, 4.351894087882593, 4.988041655057666, 5.604846211664618, 5.987982267221447, 6.420908282495581, 7.099791091246954, 7.601349626115677, 8.114249566612408, 8.506426462748760, 9.046699565852487, 9.713415430654421, 9.995170347266376, 10.57627243468486, 11.01851520965774, 11.49726381707720, 11.97817377303267, 12.63921529798052, 13.04599885275312, 13.28582320534647

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