Properties

Label 2-26520-1.1-c1-0-4
Degree $2$
Conductor $26520$
Sign $1$
Analytic cond. $211.763$
Root an. cond. $14.5520$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 2·7-s + 9-s + 13-s − 15-s + 17-s + 4·19-s − 2·21-s + 8·23-s + 25-s + 27-s + 2·29-s + 2·31-s + 2·35-s + 4·37-s + 39-s − 6·41-s + 12·43-s − 45-s − 6·47-s − 3·49-s + 51-s + 6·53-s + 4·57-s − 2·59-s − 2·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.277·13-s − 0.258·15-s + 0.242·17-s + 0.917·19-s − 0.436·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.359·31-s + 0.338·35-s + 0.657·37-s + 0.160·39-s − 0.937·41-s + 1.82·43-s − 0.149·45-s − 0.875·47-s − 3/7·49-s + 0.140·51-s + 0.824·53-s + 0.529·57-s − 0.260·59-s − 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26520\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(211.763\)
Root analytic conductor: \(14.5520\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 26520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.651378685\)
\(L(\frac12)\) \(\approx\) \(2.651378685\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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

−15.37459384102004, −14.78178571870702, −14.21866579170545, −13.72373357642452, −13.03658378228373, −12.83379373219606, −12.09500811177173, −11.54993019827311, −10.98858190620828, −10.32945151771798, −9.803812768510371, −9.145169171271912, −8.883675952449925, −8.014442419143513, −7.637600501748073, −6.912335593474231, −6.516087999032995, −5.676834914818590, −5.006141164399350, −4.334823271832925, −3.560601889517154, −3.084417190903902, −2.550221807511151, −1.381922928316845, −0.6649683820279837, 0.6649683820279837, 1.381922928316845, 2.550221807511151, 3.084417190903902, 3.560601889517154, 4.334823271832925, 5.006141164399350, 5.676834914818590, 6.516087999032995, 6.912335593474231, 7.637600501748073, 8.014442419143513, 8.883675952449925, 9.145169171271912, 9.803812768510371, 10.32945151771798, 10.98858190620828, 11.54993019827311, 12.09500811177173, 12.83379373219606, 13.03658378228373, 13.72373357642452, 14.21866579170545, 14.78178571870702, 15.37459384102004

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