Properties

Label 2-189618-1.1-c1-0-30
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 − 4·7-s − 8-s + 9-s + 2·10-s − 11-s − 12-s + 4·14-s + 2·15-s + 16-s + 17-s − 18-s − 4·19-s − 2·20-s + 4·21-s + 22-s − 4·23-s + 24-s − 25-s − 27-s − 4·28-s − 6·29-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 − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s − 0.288·12-s + 1.06·14-s + 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.872·21-s + 0.213·22-s − 0.834·23-s + 0.204·24-s − 1/5·25-s − 0.192·27-s − 0.755·28-s − 1.11·29-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: \(2\)
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 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.54514147880518, −12.96581817066655, −12.45639637640687, −12.06780360122653, −11.86735447728206, −11.08933287313832, −10.75945028641256, −10.22575236357951, −9.892714230932167, −9.396637863681827, −8.867804720261815, −8.303908106665658, −7.878684171209279, −7.302435478492502, −6.881186828185882, −6.466858833252671, −5.820662145964507, −5.591270092191746, −4.688621184882303, −3.988792267692107, −3.739802495533634, −3.060738624222584, −2.434693391121293, −1.774165614752060, −0.8537392533016001, 0, 0, 0.8537392533016001, 1.774165614752060, 2.434693391121293, 3.060738624222584, 3.739802495533634, 3.988792267692107, 4.688621184882303, 5.591270092191746, 5.820662145964507, 6.466858833252671, 6.881186828185882, 7.302435478492502, 7.878684171209279, 8.303908106665658, 8.867804720261815, 9.396637863681827, 9.892714230932167, 10.22575236357951, 10.75945028641256, 11.08933287313832, 11.86735447728206, 12.06780360122653, 12.45639637640687, 12.96581817066655, 13.54514147880518

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