Properties

Label 2-189618-1.1-c1-0-0
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 − 6-s − 2·7-s − 8-s + 9-s − 11-s + 12-s + 2·14-s + 16-s − 17-s − 18-s − 8·19-s − 2·21-s + 22-s + 6·23-s − 24-s − 5·25-s + 27-s − 2·28-s + 6·29-s + 4·31-s − 32-s − 33-s + 34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.83·19-s − 0.436·21-s + 0.213·22-s + 1.25·23-s − 0.204·24-s − 25-s + 0.192·27-s − 0.377·28-s + 1.11·29-s + 0.718·31-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: \(0\)
Selberg data: \((2,\ 189618,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3734297343\)
\(L(\frac12)\) \(\approx\) \(0.3734297343\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−13.10812059123837, −12.66517341897559, −12.30217835405010, −11.62019118066289, −11.00841614163178, −10.79568668306524, −10.08786333369465, −9.789108977795229, −9.398760549047118, −8.691984693700203, −8.447436108113492, −8.043369121359644, −7.395610158669385, −6.821936665632323, −6.494847404017912, −6.058782218342831, −5.343073321937034, −4.540766826737226, −4.267915438155274, −3.450897558709752, −2.823512475124256, −2.629287798101796, −1.754401458671713, −1.262979740866259, −0.1850041630514780, 0.1850041630514780, 1.262979740866259, 1.754401458671713, 2.629287798101796, 2.823512475124256, 3.450897558709752, 4.267915438155274, 4.540766826737226, 5.343073321937034, 6.058782218342831, 6.494847404017912, 6.821936665632323, 7.395610158669385, 8.043369121359644, 8.447436108113492, 8.691984693700203, 9.398760549047118, 9.789108977795229, 10.08786333369465, 10.79568668306524, 11.00841614163178, 11.62019118066289, 12.30217835405010, 12.66517341897559, 13.10812059123837

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