Properties

Label 2-189618-1.1-c1-0-27
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 + 6-s + 4·7-s − 8-s + 9-s − 11-s − 12-s − 4·14-s + 16-s − 17-s − 18-s − 2·19-s − 4·21-s + 22-s + 4·23-s + 24-s − 5·25-s − 27-s + 4·28-s − 2·29-s − 32-s + 33-s + 34-s + 36-s − 10·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.458·19-s − 0.872·21-s + 0.213·22-s + 0.834·23-s + 0.204·24-s − 25-s − 0.192·27-s + 0.755·28-s − 0.371·29-s − 0.176·32-s + 0.174·33-s + 0.171·34-s + 1/6·36-s − 1.64·37-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 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.23687837223750, −12.88725923435880, −12.18435673891233, −11.72155595815780, −11.48596754317197, −10.87234319401653, −10.73780386735790, −10.01424901324134, −9.733590059731773, −8.928392036271688, −8.542392663719077, −8.195195078898991, −7.615214477022346, −7.176149696183234, −6.712862771593402, −6.111222026977765, −5.436017403562074, −5.121055480443338, −4.655840094893072, −3.922065655032105, −3.417491356647094, −2.462846956020116, −1.954234758510239, −1.519328406825256, −0.7698938798955691, 0, 0.7698938798955691, 1.519328406825256, 1.954234758510239, 2.462846956020116, 3.417491356647094, 3.922065655032105, 4.655840094893072, 5.121055480443338, 5.436017403562074, 6.111222026977765, 6.712862771593402, 7.176149696183234, 7.615214477022346, 8.195195078898991, 8.542392663719077, 8.928392036271688, 9.733590059731773, 10.01424901324134, 10.73780386735790, 10.87234319401653, 11.48596754317197, 11.72155595815780, 12.18435673891233, 12.88725923435880, 13.23687837223750

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