Properties

Label 2-47190-1.1-c1-0-39
Degree $2$
Conductor $47190$
Sign $-1$
Analytic cond. $376.814$
Root an. cond. $19.4116$
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 − 5-s − 6-s − 3·7-s − 8-s + 9-s + 10-s + 12-s + 13-s + 3·14-s − 15-s + 16-s + 6·17-s − 18-s + 3·19-s − 20-s − 3·21-s + 4·23-s − 24-s + 25-s − 26-s + 27-s − 3·28-s + 2·29-s + 30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 0.277·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.688·19-s − 0.223·20-s − 0.654·21-s + 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.566·28-s + 0.371·29-s + 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(47190\)    =    \(2 \cdot 3 \cdot 5 \cdot 11^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(376.814\)
Root analytic conductor: \(19.4116\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{47190} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 47190,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 + T \)
11 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−15.12904030534786, −14.29017308351667, −13.83736871112252, −13.38414235585725, −12.65839343060265, −12.21218007245451, −11.90534759318903, −11.12011255967144, −10.51524235836190, −10.04486299440972, −9.616258128956582, −9.152332290551537, −8.477781512867913, −8.115830307979058, −7.464626309061020, −6.968293068578033, −6.515596064112379, −5.777993533909052, −5.164517100975139, −4.341766776398973, −3.480615404736271, −3.155610614251765, −2.718959503334829, −1.552533843690782, −0.9838998185931907, 0, 0.9838998185931907, 1.552533843690782, 2.718959503334829, 3.155610614251765, 3.480615404736271, 4.341766776398973, 5.164517100975139, 5.777993533909052, 6.515596064112379, 6.968293068578033, 7.464626309061020, 8.115830307979058, 8.477781512867913, 9.152332290551537, 9.616258128956582, 10.04486299440972, 10.51524235836190, 11.12011255967144, 11.90534759318903, 12.21218007245451, 12.65839343060265, 13.38414235585725, 13.83736871112252, 14.29017308351667, 15.12904030534786

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