Properties

Label 2-189618-1.1-c1-0-1
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 − 4·5-s − 6-s + 2·7-s − 8-s + 9-s + 4·10-s − 11-s + 12-s − 2·14-s − 4·15-s + 16-s − 17-s − 18-s − 4·20-s + 2·21-s + 22-s − 6·23-s − 24-s + 11·25-s + 27-s + 2·28-s − 2·29-s + 4·30-s − 4·31-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.301·11-s + 0.288·12-s − 0.534·14-s − 1.03·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.894·20-s + 0.436·21-s + 0.213·22-s − 1.25·23-s − 0.204·24-s + 11/5·25-s + 0.192·27-s + 0.377·28-s − 0.371·29-s + 0.730·30-s − 0.718·31-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

\(L(1)\) \(\approx\) \(0.7077851056\)
\(L(\frac12)\) \(\approx\) \(0.7077851056\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−12.97303454978115, −12.45991993520033, −12.13530105610746, −11.55255089136676, −11.31424846712103, −10.72296500225335, −10.42446286065043, −9.747924437462895, −9.160531734162231, −8.599845448241798, −8.437937255430236, −7.788961190394617, −7.492405465246648, −7.308974434225179, −6.496551739148740, −5.915008229217520, −5.159391887891432, −4.588764430803640, −4.110258642894278, −3.625831253189206, −3.116513148585275, −2.376345231331114, −1.836341925552410, −1.079627611934610, −0.2851302531404390, 0.2851302531404390, 1.079627611934610, 1.836341925552410, 2.376345231331114, 3.116513148585275, 3.625831253189206, 4.110258642894278, 4.588764430803640, 5.159391887891432, 5.915008229217520, 6.496551739148740, 7.308974434225179, 7.492405465246648, 7.788961190394617, 8.437937255430236, 8.599845448241798, 9.160531734162231, 9.747924437462895, 10.42446286065043, 10.72296500225335, 11.31424846712103, 11.55255089136676, 12.13530105610746, 12.45991993520033, 12.97303454978115

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