Properties

Label 2-189618-1.1-c1-0-24
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 − 8-s + 9-s − 2·10-s − 11-s − 12-s − 2·15-s + 16-s − 17-s − 18-s + 4·19-s + 2·20-s + 22-s − 4·23-s + 24-s − 25-s − 27-s + 2·30-s − 2·31-s − 32-s + 33-s + 34-s + 36-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 + 0.917·19-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 + 0.365·30-s − 0.359·31-s − 0.176·32-s + 0.174·33-s + 0.171·34-s + 1/6·36-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 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−13.41474985844287, −12.75448464103567, −12.38362591082271, −11.82832886998992, −11.36478188010124, −11.04589051821943, −10.32073109610885, −9.953606319599769, −9.830002113359317, −9.094228742084883, −8.740014041131913, −8.006457244607499, −7.686613327895664, −7.079515713653658, −6.526514802539746, −6.145656308739706, −5.653125228849999, −5.090306839570642, −4.744417823825582, −3.771706147771581, −3.351290495690680, −2.559880452415753, −1.918858419503089, −1.581639364973890, −0.7312405146227224, 0, 0.7312405146227224, 1.581639364973890, 1.918858419503089, 2.559880452415753, 3.351290495690680, 3.771706147771581, 4.744417823825582, 5.090306839570642, 5.653125228849999, 6.145656308739706, 6.526514802539746, 7.079515713653658, 7.686613327895664, 8.006457244607499, 8.740014041131913, 9.094228742084883, 9.830002113359317, 9.953606319599769, 10.32073109610885, 11.04589051821943, 11.36478188010124, 11.82832886998992, 12.38362591082271, 12.75448464103567, 13.41474985844287

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