Properties

Label 2-17160-1.1-c1-0-1
Degree $2$
Conductor $17160$
Sign $1$
Analytic cond. $137.023$
Root an. cond. $11.7056$
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 − 4·7-s + 9-s − 11-s + 13-s + 15-s − 2·17-s + 4·19-s + 4·21-s − 4·23-s + 25-s − 27-s + 6·29-s + 33-s + 4·35-s + 10·37-s − 39-s − 10·41-s + 4·43-s − 45-s + 9·49-s + 2·51-s + 10·53-s + 55-s − 4·57-s − 12·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 0.258·15-s − 0.485·17-s + 0.917·19-s + 0.872·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.174·33-s + 0.676·35-s + 1.64·37-s − 0.160·39-s − 1.56·41-s + 0.609·43-s − 0.149·45-s + 9/7·49-s + 0.280·51-s + 1.37·53-s + 0.134·55-s − 0.529·57-s − 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17160\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 13\)
Sign: $1$
Analytic conductor: \(137.023\)
Root analytic conductor: \(11.7056\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{17160} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 17160,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6747888862\)
\(L(\frac12)\) \(\approx\) \(0.6747888862\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−15.99897628785129, −15.46823596350373, −15.02225232371058, −14.02377407158622, −13.59891100894075, −13.07187181820385, −12.46174357998330, −12.06121831475583, −11.45233318362287, −10.87134763337271, −10.10362647295153, −9.861078375148711, −9.171546990115352, −8.437975370013624, −7.824682579514006, −6.988558621651079, −6.700842257414838, −5.897502855074067, −5.528333676553616, −4.462701157342939, −4.044510787285396, −3.103597667149584, −2.685942815458907, −1.376090895638587, −0.3768695254336265, 0.3768695254336265, 1.376090895638587, 2.685942815458907, 3.103597667149584, 4.044510787285396, 4.462701157342939, 5.528333676553616, 5.897502855074067, 6.700842257414838, 6.988558621651079, 7.824682579514006, 8.437975370013624, 9.171546990115352, 9.861078375148711, 10.10362647295153, 10.87134763337271, 11.45233318362287, 12.06121831475583, 12.46174357998330, 13.07187181820385, 13.59891100894075, 14.02377407158622, 15.02225232371058, 15.46823596350373, 15.99897628785129

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