Properties

Label 2-26520-1.1-c1-0-23
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

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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 + 25-s + 27-s + 6·29-s − 10·31-s − 2·35-s − 4·37-s + 39-s − 2·41-s + 4·43-s + 45-s − 6·47-s − 3·49-s − 51-s − 6·53-s + 4·57-s − 6·59-s + 6·61-s − 2·63-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 + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.79·31-s − 0.338·35-s − 0.657·37-s + 0.160·39-s − 0.312·41-s + 0.609·43-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 − 0.781·59-s + 0.768·61-s − 0.251·63-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.62034046341440, −14.99907296369228, −14.32470688132399, −14.06897326605110, −13.33187862241255, −13.07969860538222, −12.42022840280306, −11.94479932105054, −11.11691718180344, −10.67643612185310, −9.929058291222910, −9.614711788353702, −8.993774814558035, −8.584188337792284, −7.760126052980759, −7.290325401891593, −6.560009794051454, −6.166877889910977, −5.324610273185608, −4.818595381021835, −3.870804169455698, −3.335166083914896, −2.776036595443703, −1.923594591935979, −1.195162785633900, 0, 1.195162785633900, 1.923594591935979, 2.776036595443703, 3.335166083914896, 3.870804169455698, 4.818595381021835, 5.324610273185608, 6.166877889910977, 6.560009794051454, 7.290325401891593, 7.760126052980759, 8.584188337792284, 8.993774814558035, 9.614711788353702, 9.929058291222910, 10.67643612185310, 11.11691718180344, 11.94479932105054, 12.42022840280306, 13.07969860538222, 13.33187862241255, 14.06897326605110, 14.32470688132399, 14.99907296369228, 15.62034046341440

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