Properties

Label 2-15210-1.1-c1-0-0
Degree $2$
Conductor $15210$
Sign $1$
Analytic cond. $121.452$
Root an. cond. $11.0205$
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  − 2-s + 4-s − 5-s − 2·7-s − 8-s + 10-s − 3·11-s + 2·14-s + 16-s − 6·17-s − 2·19-s − 20-s + 3·22-s − 3·23-s + 25-s − 2·28-s − 3·29-s − 5·31-s − 32-s + 6·34-s + 2·35-s + 7·37-s + 2·38-s + 40-s + 6·41-s − 43-s − 3·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s − 0.904·11-s + 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.458·19-s − 0.223·20-s + 0.639·22-s − 0.625·23-s + 1/5·25-s − 0.377·28-s − 0.557·29-s − 0.898·31-s − 0.176·32-s + 1.02·34-s + 0.338·35-s + 1.15·37-s + 0.324·38-s + 0.158·40-s + 0.937·41-s − 0.152·43-s − 0.452·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15210\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(121.452\)
Root analytic conductor: \(11.0205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 15210,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1416209176\)
\(L(\frac12)\) \(\approx\) \(0.1416209176\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
13 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−16.07883120421885, −15.69735331024790, −14.97794502141233, −14.68522140714199, −13.62029970230937, −13.16114672615271, −12.72785289106122, −12.09780033123057, −11.26384709131970, −11.00865314931735, −10.33229789469418, −9.758571375061680, −9.121350822021680, −8.645903780260878, −7.933571607693251, −7.415329422619561, −6.808328113393267, −6.122268488888528, −5.582068002241290, −4.520720335335242, −4.020659176736674, −3.007618351420407, −2.485132027382350, −1.572629730444963, −0.1758368699353276, 0.1758368699353276, 1.572629730444963, 2.485132027382350, 3.007618351420407, 4.020659176736674, 4.520720335335242, 5.582068002241290, 6.122268488888528, 6.808328113393267, 7.415329422619561, 7.933571607693251, 8.645903780260878, 9.121350822021680, 9.758571375061680, 10.33229789469418, 11.00865314931735, 11.26384709131970, 12.09780033123057, 12.72785289106122, 13.16114672615271, 13.62029970230937, 14.68522140714199, 14.97794502141233, 15.69735331024790, 16.07883120421885

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