Properties

Label 2-26520-1.1-c1-0-3
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s + 4·11-s + 13-s − 15-s + 17-s − 8·19-s − 8·23-s + 25-s + 27-s + 6·29-s + 4·33-s − 6·37-s + 39-s + 6·41-s + 4·43-s − 45-s + 4·47-s − 7·49-s + 51-s − 10·53-s − 4·55-s − 8·57-s + 6·61-s − 65-s − 8·69-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 + 0.242·17-s − 1.83·19-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.696·33-s − 0.986·37-s + 0.160·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s + 0.583·47-s − 49-s + 0.140·51-s − 1.37·53-s − 0.539·55-s − 1.05·57-s + 0.768·61-s − 0.124·65-s − 0.963·69-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: \(0\)
Selberg data: \((2,\ 26520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.415606272\)
\(L(\frac12)\) \(\approx\) \(2.415606272\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.33162297916699, −14.57275109340675, −14.28753101783345, −13.94273451426500, −13.09649399337437, −12.58001857485359, −12.13274870972643, −11.62835071711745, −10.89963117390699, −10.43545510093207, −9.807342700282696, −9.181474761271823, −8.643507865577362, −8.194715073928983, −7.686662946745351, −6.856692652060138, −6.365227930673726, −5.931715308778547, −4.824932986731635, −4.212167255953597, −3.856249345758655, −3.130014156678169, −2.207796173120936, −1.637490171263562, −0.5877237924917512, 0.5877237924917512, 1.637490171263562, 2.207796173120936, 3.130014156678169, 3.856249345758655, 4.212167255953597, 4.824932986731635, 5.931715308778547, 6.365227930673726, 6.856692652060138, 7.686662946745351, 8.194715073928983, 8.643507865577362, 9.181474761271823, 9.807342700282696, 10.43545510093207, 10.89963117390699, 11.62835071711745, 12.13274870972643, 12.58001857485359, 13.09649399337437, 13.94273451426500, 14.28753101783345, 14.57275109340675, 15.33162297916699

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