Properties

Label 2-15e2-1.1-c3-0-19
Degree $2$
Conductor $225$
Sign $-1$
Analytic cond. $13.2754$
Root an. cond. $3.64354$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 4-s − 20·7-s − 21·8-s + 24·11-s − 74·13-s − 60·14-s − 71·16-s + 54·17-s − 124·19-s + 72·22-s − 120·23-s − 222·26-s − 20·28-s + 78·29-s + 200·31-s − 45·32-s + 162·34-s + 70·37-s − 372·38-s − 330·41-s − 92·43-s + 24·44-s − 360·46-s − 24·47-s + 57·49-s − 74·52-s + ⋯
L(s)  = 1  + 1.06·2-s + 1/8·4-s − 1.07·7-s − 0.928·8-s + 0.657·11-s − 1.57·13-s − 1.14·14-s − 1.10·16-s + 0.770·17-s − 1.49·19-s + 0.697·22-s − 1.08·23-s − 1.67·26-s − 0.134·28-s + 0.499·29-s + 1.15·31-s − 0.248·32-s + 0.817·34-s + 0.311·37-s − 1.58·38-s − 1.25·41-s − 0.326·43-s + 0.0822·44-s − 1.15·46-s − 0.0744·47-s + 0.166·49-s − 0.197·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 - 3 T + p^{3} T^{2} \)
7 \( 1 + 20 T + p^{3} T^{2} \)
11 \( 1 - 24 T + p^{3} T^{2} \)
13 \( 1 + 74 T + p^{3} T^{2} \)
17 \( 1 - 54 T + p^{3} T^{2} \)
19 \( 1 + 124 T + p^{3} T^{2} \)
23 \( 1 + 120 T + p^{3} T^{2} \)
29 \( 1 - 78 T + p^{3} T^{2} \)
31 \( 1 - 200 T + p^{3} T^{2} \)
37 \( 1 - 70 T + p^{3} T^{2} \)
41 \( 1 + 330 T + p^{3} T^{2} \)
43 \( 1 + 92 T + p^{3} T^{2} \)
47 \( 1 + 24 T + p^{3} T^{2} \)
53 \( 1 - 450 T + p^{3} T^{2} \)
59 \( 1 + 24 T + p^{3} T^{2} \)
61 \( 1 + 322 T + p^{3} T^{2} \)
67 \( 1 - 196 T + p^{3} T^{2} \)
71 \( 1 - 288 T + p^{3} T^{2} \)
73 \( 1 - 430 T + p^{3} T^{2} \)
79 \( 1 + 520 T + p^{3} T^{2} \)
83 \( 1 - 156 T + p^{3} T^{2} \)
89 \( 1 + 1026 T + p^{3} T^{2} \)
97 \( 1 - 286 T + p^{3} T^{2} \)
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

−11.90778819164220862100965240576, −10.19144311086461557443487565667, −9.559974877637067924580253906881, −8.348596345906738696540401957759, −6.82730598190250880689978538633, −6.06613246028294637687707169883, −4.80750741579903131949499602018, −3.78677782881581625327031002249, −2.57978758470335437435358267536, 0, 2.57978758470335437435358267536, 3.78677782881581625327031002249, 4.80750741579903131949499602018, 6.06613246028294637687707169883, 6.82730598190250880689978538633, 8.348596345906738696540401957759, 9.559974877637067924580253906881, 10.19144311086461557443487565667, 11.90778819164220862100965240576

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