Properties

Label 2-189618-1.1-c1-0-10
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 + 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 + 6·19-s + 2·20-s + 22-s + 4·23-s + 24-s − 25-s − 27-s − 6·29-s + 2·30-s + 8·31-s − 32-s + 33-s − 34-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 + 1.37·19-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 − 1.11·29-s + 0.365·30-s + 1.43·31-s − 0.176·32-s + 0.174·33-s − 0.171·34-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

\(L(1)\) \(\approx\) \(1.477470826\)
\(L(\frac12)\) \(\approx\) \(1.477470826\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 5 T + p T^{2} \)
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

−13.25042535801673, −12.55827479369371, −12.05047482025122, −11.68430168533635, −10.99942730429238, −10.89733421226277, −10.17633209042023, −9.735539414735428, −9.386285040205665, −9.172796577861158, −8.179696484052011, −7.891410493398157, −7.437866157664414, −6.766610017133585, −6.372020223379230, −5.808102107927226, −5.476776780079745, −4.855701791831837, −4.383455110643497, −3.409693969801514, −2.982018573566980, −2.383604059852546, −1.541061409957248, −1.262238251648070, −0.4201099038408691, 0.4201099038408691, 1.262238251648070, 1.541061409957248, 2.383604059852546, 2.982018573566980, 3.409693969801514, 4.383455110643497, 4.855701791831837, 5.476776780079745, 5.808102107927226, 6.372020223379230, 6.766610017133585, 7.437866157664414, 7.891410493398157, 8.179696484052011, 9.172796577861158, 9.386285040205665, 9.735539414735428, 10.17633209042023, 10.89733421226277, 10.99942730429238, 11.68430168533635, 12.05047482025122, 12.55827479369371, 13.25042535801673

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