Properties

Label 2-26520-1.1-c1-0-12
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 $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 4·7-s + 9-s + 4·11-s − 13-s − 15-s + 17-s − 4·19-s − 4·21-s − 8·23-s + 25-s + 27-s + 2·29-s + 4·33-s + 4·35-s + 2·37-s − 39-s − 2·41-s + 8·43-s − 45-s + 9·49-s + 51-s + 10·53-s − 4·55-s − 4·57-s − 4·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.258·15-s + 0.242·17-s − 0.917·19-s − 0.872·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.696·33-s + 0.676·35-s + 0.328·37-s − 0.160·39-s − 0.312·41-s + 1.21·43-s − 0.149·45-s + 9/7·49-s + 0.140·51-s + 1.37·53-s − 0.539·55-s − 0.529·57-s − 0.520·59-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: $\chi_{26520} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 26520,\ (\ :1/2),\ -1)\)

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

−15.62767118571845, −14.94963004760301, −14.56362580964380, −13.91353299877258, −13.54066305949330, −12.77918498953838, −12.35886093364579, −12.03252038991881, −11.31021502232848, −10.51436280344784, −10.05826493188501, −9.443762401956700, −9.167598407859801, −8.360820220818872, −7.966178578779436, −7.089051059886441, −6.686065073712519, −6.174316099709422, −5.527717850112004, −4.386091681060709, −3.956684249002865, −3.534849623290952, −2.693831398420031, −2.081059577505970, −0.9703469537705693, 0, 0.9703469537705693, 2.081059577505970, 2.693831398420031, 3.534849623290952, 3.956684249002865, 4.386091681060709, 5.527717850112004, 6.174316099709422, 6.686065073712519, 7.089051059886441, 7.966178578779436, 8.360820220818872, 9.167598407859801, 9.443762401956700, 10.05826493188501, 10.51436280344784, 11.31021502232848, 12.03252038991881, 12.35886093364579, 12.77918498953838, 13.54066305949330, 13.91353299877258, 14.56362580964380, 14.94963004760301, 15.62767118571845

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