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Properties

Label 2-3724-19.18-c0-0-2
Degree $2$
Conductor $3724$
Sign $1$
Analytic cond. $1.85851$
Root an. cond. $1.36327$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 9-s − 11-s + 17-s − 19-s + 2·23-s − 43-s + 45-s + 47-s − 55-s + 61-s + 73-s + 81-s − 2·83-s + 85-s − 95-s − 99-s − 2·101-s + 2·115-s + ⋯
L(s)  = 1  + 5-s + 9-s − 11-s + 17-s − 19-s + 2·23-s − 43-s + 45-s + 47-s − 55-s + 61-s + 73-s + 81-s − 2·83-s + 85-s − 95-s − 99-s − 2·101-s + 2·115-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(1.85851\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3724} (1177, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.645931600\)
\(L(\frac12)\) \(\approx\) \(1.645931600\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 + T \)
good3 \( ( 1 - T )( 1 + T ) \)
5 \( 1 - T + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 - T + T^{2} \)
23 \( ( 1 - T )^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T + T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 - T + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 - T + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 + T )^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−8.739069804882857071314193889888, −7.941512745255485094013409165866, −7.14571693750860645095381017698, −6.55651586654537790333099172848, −5.56313536195293300891189663434, −5.10676978806591527363989276293, −4.16192034255348527521532265532, −3.07187770839134501258703156915, −2.20371899109842608919718415978, −1.21333222347854062807095710307, 1.21333222347854062807095710307, 2.20371899109842608919718415978, 3.07187770839134501258703156915, 4.16192034255348527521532265532, 5.10676978806591527363989276293, 5.56313536195293300891189663434, 6.55651586654537790333099172848, 7.14571693750860645095381017698, 7.941512745255485094013409165866, 8.739069804882857071314193889888

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