Properties

Label 2-15210-1.1-c1-0-4
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s − 4·7-s + 8-s − 10-s − 5·11-s − 4·14-s + 16-s + 2·17-s − 6·19-s − 20-s − 5·22-s − 9·23-s + 25-s − 4·28-s + 5·29-s − 3·31-s + 32-s + 2·34-s + 4·35-s − 5·37-s − 6·38-s − 40-s − 2·41-s − 7·43-s − 5·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s + 0.353·8-s − 0.316·10-s − 1.50·11-s − 1.06·14-s + 1/4·16-s + 0.485·17-s − 1.37·19-s − 0.223·20-s − 1.06·22-s − 1.87·23-s + 1/5·25-s − 0.755·28-s + 0.928·29-s − 0.538·31-s + 0.176·32-s + 0.342·34-s + 0.676·35-s − 0.821·37-s − 0.973·38-s − 0.158·40-s − 0.312·41-s − 1.06·43-s − 0.753·44-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

\(L(1)\) \(\approx\) \(0.7614141963\)
\(L(\frac12)\) \(\approx\) \(0.7614141963\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.96067145353256, −15.46133746163946, −15.13379873331604, −14.28203018844961, −13.66132044857033, −13.27854794298512, −12.61872904467576, −12.30526146436108, −11.83102196790923, −10.79008836172645, −10.38901230590262, −10.06796564821511, −9.215324890687755, −8.392320330216391, −7.904127912037832, −7.233713228450440, −6.503503733881863, −6.065312196648747, −5.404698374462544, −4.625003277107271, −3.902322863182593, −3.306435503756329, −2.658648184923310, −1.919812587091756, −0.3116915734562921, 0.3116915734562921, 1.919812587091756, 2.658648184923310, 3.306435503756329, 3.902322863182593, 4.625003277107271, 5.404698374462544, 6.065312196648747, 6.503503733881863, 7.233713228450440, 7.904127912037832, 8.392320330216391, 9.215324890687755, 10.06796564821511, 10.38901230590262, 10.79008836172645, 11.83102196790923, 12.30526146436108, 12.61872904467576, 13.27854794298512, 13.66132044857033, 14.28203018844961, 15.13379873331604, 15.46133746163946, 15.96067145353256

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