Properties

Label 2-26520-1.1-c1-0-31
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 $2$

Origins

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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 − 6·29-s − 8·31-s − 4·33-s + 4·35-s − 2·37-s + 39-s + 6·41-s − 4·43-s − 45-s + 9·49-s − 51-s − 6·53-s + 4·55-s − 4·57-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 − 1.11·29-s − 1.43·31-s − 0.696·33-s + 0.676·35-s − 0.328·37-s + 0.160·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s + 9/7·49-s − 0.140·51-s − 0.824·53-s + 0.539·55-s − 0.529·57-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: \(2\)
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 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−15.91817405151766, −15.32449881403410, −14.89052007862327, −14.19930800378621, −13.57229780165767, −13.12238855216221, −12.68539512638270, −12.36130777714039, −11.52102523906825, −10.74582184202266, −10.47559599699639, −9.789553526812361, −9.251002529378206, −8.806494640918883, −7.965044876210844, −7.684283749596563, −7.004399722841802, −6.256393800754498, −5.884131316727082, −5.043323484247466, −4.142779950152997, −3.713488777856412, −3.077670002545477, −2.408424288327939, −1.684884406822399, 0, 0, 1.684884406822399, 2.408424288327939, 3.077670002545477, 3.713488777856412, 4.142779950152997, 5.043323484247466, 5.884131316727082, 6.256393800754498, 7.004399722841802, 7.684283749596563, 7.965044876210844, 8.806494640918883, 9.251002529378206, 9.789553526812361, 10.47559599699639, 10.74582184202266, 11.52102523906825, 12.36130777714039, 12.68539512638270, 13.12238855216221, 13.57229780165767, 14.19930800378621, 14.89052007862327, 15.32449881403410, 15.91817405151766

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