↑

Properties

Label 2-76440-1.1-c1-0-71
Degree $2$
Conductor $76440$
Sign $-1$
Analytic cond. $610.376$
Root an. cond. $24.7057$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 9-s + 4·11-s + 13-s − 15-s + 2·19-s + 8·23-s + 25-s − 27-s − 2·29-s − 2·31-s − 4·33-s − 8·37-s − 39-s + 6·41-s − 10·43-s + 45-s − 2·47-s + 2·53-s + 4·55-s − 2·57-s − 8·59-s + 65-s + 4·67-s − 8·69-s − 10·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.258·15-s + 0.458·19-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.359·31-s − 0.696·33-s − 1.31·37-s − 0.160·39-s + 0.937·41-s − 1.52·43-s + 0.149·45-s − 0.291·47-s + 0.274·53-s + 0.539·55-s − 0.264·57-s − 1.04·59-s + 0.124·65-s + 0.488·67-s − 0.963·69-s − 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76440\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(610.376\)
Root analytic conductor: \(24.7057\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 76440,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 \)
13 \( 1 - T \)
good11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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

−14.43445548367247, −13.68967507922818, −13.30127077127416, −12.81777423009352, −12.25515895590699, −11.75442013543334, −11.32490343189367, −10.87036558943030, −10.30704023746238, −9.795968884867470, −9.094665086164588, −8.997357321094214, −8.276640829121425, −7.421515684250439, −7.057498492857447, −6.524241022695003, −6.052231461346458, −5.421852542755683, −4.942712044291015, −4.378139790202662, −3.555151362260732, −3.217408854746152, −2.254802023288888, −1.438158102380821, −1.099415684498600, 0, 1.099415684498600, 1.438158102380821, 2.254802023288888, 3.217408854746152, 3.555151362260732, 4.378139790202662, 4.942712044291015, 5.421852542755683, 6.052231461346458, 6.524241022695003, 7.057498492857447, 7.421515684250439, 8.276640829121425, 8.997357321094214, 9.094665086164588, 9.795968884867470, 10.30704023746238, 10.87036558943030, 11.32490343189367, 11.75442013543334, 12.25515895590699, 12.81777423009352, 13.30127077127416, 13.68967507922818, 14.43445548367247

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