Properties

Label 2-15210-1.1-c1-0-16
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 − 7-s + 8-s + 10-s − 3·11-s − 14-s + 16-s + 6·17-s + 5·19-s + 20-s − 3·22-s + 25-s − 28-s − 4·31-s + 32-s + 6·34-s − 35-s + 11·37-s + 5·38-s + 40-s − 6·41-s + 2·43-s − 3·44-s + 3·47-s − 6·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s − 0.904·11-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 1.14·19-s + 0.223·20-s − 0.639·22-s + 1/5·25-s − 0.188·28-s − 0.718·31-s + 0.176·32-s + 1.02·34-s − 0.169·35-s + 1.80·37-s + 0.811·38-s + 0.158·40-s − 0.937·41-s + 0.304·43-s − 0.452·44-s + 0.437·47-s − 6/7·49-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

\(L(1)\) \(\approx\) \(3.853527787\)
\(L(\frac12)\) \(\approx\) \(3.853527787\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−16.07428609941936, −15.39465182029802, −14.88034252711243, −14.27883086698053, −13.84053532847809, −13.15835774632534, −12.87312778741435, −12.20000357724200, −11.62928751503201, −11.07217662558596, −10.18806050239044, −10.01001311453326, −9.321085118078369, −8.501768091510699, −7.661119887583280, −7.419829040884092, −6.560615856414053, −5.728187706795610, −5.531071524500611, −4.805530765421424, −3.934583782200110, −3.142445901889181, −2.735372393743836, −1.727332550132964, −0.7743178674447364, 0.7743178674447364, 1.727332550132964, 2.735372393743836, 3.142445901889181, 3.934583782200110, 4.805530765421424, 5.531071524500611, 5.728187706795610, 6.560615856414053, 7.419829040884092, 7.661119887583280, 8.501768091510699, 9.321085118078369, 10.01001311453326, 10.18806050239044, 11.07217662558596, 11.62928751503201, 12.20000357724200, 12.87312778741435, 13.15835774632534, 13.84053532847809, 14.27883086698053, 14.88034252711243, 15.39465182029802, 16.07428609941936

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