↑

Properties

Label 2-3724-19.18-c0-0-4
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

Downloads

Learn more

Normalization:  

Dirichlet series

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

\(L(\frac{1}{2})\) \(\approx\) \(1.846985638\)
\(L(\frac12)\) \(\approx\) \(1.846985638\)
\(L(1)\) 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 \( ( 1 - T )^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 - T + T^{2} \)
23 \( 1 + T + T^{2} \)
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 \( ( 1 + T )^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 + T )^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 + T )^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 - T + T^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.834143490929694070915717610039, −7.978613340180315120884266922031, −7.08167270764593838438470743214, −6.31929533684378567965287081919, −6.02580187154272229622696660458, −4.83150596569342014086982645391, −4.10276045405290476962344075868, −3.32805090367175768737898483891, −1.86411914501531901952158312519, −1.44188208273219707419777649227, 1.44188208273219707419777649227, 1.86411914501531901952158312519, 3.32805090367175768737898483891, 4.10276045405290476962344075868, 4.83150596569342014086982645391, 6.02580187154272229622696660458, 6.31929533684378567965287081919, 7.08167270764593838438470743214, 7.978613340180315120884266922031, 8.834143490929694070915717610039

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