Properties

Label 2-67760-1.1-c1-0-53
Degree $2$
Conductor $67760$
Sign $-1$
Analytic cond. $541.066$
Root an. cond. $23.2608$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 5-s − 7-s + 6·9-s + 3·13-s + 3·15-s + 17-s + 6·19-s + 3·21-s − 6·23-s + 25-s − 9·27-s + 9·29-s + 4·31-s + 35-s + 2·37-s − 9·39-s + 4·41-s + 10·43-s − 6·45-s + 47-s + 49-s − 3·51-s + 4·53-s − 18·57-s + 8·59-s + 8·61-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.447·5-s − 0.377·7-s + 2·9-s + 0.832·13-s + 0.774·15-s + 0.242·17-s + 1.37·19-s + 0.654·21-s − 1.25·23-s + 1/5·25-s − 1.73·27-s + 1.67·29-s + 0.718·31-s + 0.169·35-s + 0.328·37-s − 1.44·39-s + 0.624·41-s + 1.52·43-s − 0.894·45-s + 0.145·47-s + 1/7·49-s − 0.420·51-s + 0.549·53-s − 2.38·57-s + 1.04·59-s + 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(67760\)    =    \(2^{4} \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(541.066\)
Root analytic conductor: \(23.2608\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{67760} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 67760,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 \)
good3 \( 1 + p T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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

−14.30435539861844, −13.85679315030498, −13.35466169232878, −12.73644397135482, −12.16171910258482, −11.93797710050588, −11.54432510642474, −10.93471795148507, −10.48085332252764, −10.03561544931463, −9.567601051223175, −8.832891820594666, −8.139317539481562, −7.630251028153458, −7.037878653143269, −6.494753718450833, −6.022144720873075, −5.597852982604246, −5.055476756225529, −4.253912622375719, −4.034885886463491, −3.139779460092931, −2.411505266681509, −1.110323420398137, −0.9923063619578272, 0, 0.9923063619578272, 1.110323420398137, 2.411505266681509, 3.139779460092931, 4.034885886463491, 4.253912622375719, 5.055476756225529, 5.597852982604246, 6.022144720873075, 6.494753718450833, 7.037878653143269, 7.630251028153458, 8.139317539481562, 8.832891820594666, 9.567601051223175, 10.03561544931463, 10.48085332252764, 10.93471795148507, 11.54432510642474, 11.93797710050588, 12.16171910258482, 12.73644397135482, 13.35466169232878, 13.85679315030498, 14.30435539861844

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