Properties

Label 2-15e2-1.1-c3-0-14
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 7·4-s + 6·7-s + 15·8-s + 43·11-s − 28·13-s − 6·14-s + 41·16-s − 91·17-s − 35·19-s − 43·22-s − 162·23-s + 28·26-s − 42·28-s − 160·29-s + 42·31-s − 161·32-s + 91·34-s − 314·37-s + 35·38-s + 203·41-s + 92·43-s − 301·44-s + 162·46-s − 196·47-s − 307·49-s + 196·52-s + ⋯
L(s)  = 1  − 0.353·2-s − 7/8·4-s + 0.323·7-s + 0.662·8-s + 1.17·11-s − 0.597·13-s − 0.114·14-s + 0.640·16-s − 1.29·17-s − 0.422·19-s − 0.416·22-s − 1.46·23-s + 0.211·26-s − 0.283·28-s − 1.02·29-s + 0.243·31-s − 0.889·32-s + 0.459·34-s − 1.39·37-s + 0.149·38-s + 0.773·41-s + 0.326·43-s − 1.03·44-s + 0.519·46-s − 0.608·47-s − 0.895·49-s + 0.522·52-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 \( 1 + T + p^{3} T^{2} \)
7 \( 1 - 6 T + p^{3} T^{2} \)
11 \( 1 - 43 T + p^{3} T^{2} \)
13 \( 1 + 28 T + p^{3} T^{2} \)
17 \( 1 + 91 T + p^{3} T^{2} \)
19 \( 1 + 35 T + p^{3} T^{2} \)
23 \( 1 + 162 T + p^{3} T^{2} \)
29 \( 1 + 160 T + p^{3} T^{2} \)
31 \( 1 - 42 T + p^{3} T^{2} \)
37 \( 1 + 314 T + p^{3} T^{2} \)
41 \( 1 - 203 T + p^{3} T^{2} \)
43 \( 1 - 92 T + p^{3} T^{2} \)
47 \( 1 + 196 T + p^{3} T^{2} \)
53 \( 1 + 82 T + p^{3} T^{2} \)
59 \( 1 - 280 T + p^{3} T^{2} \)
61 \( 1 + 518 T + p^{3} T^{2} \)
67 \( 1 - 141 T + p^{3} T^{2} \)
71 \( 1 + 412 T + p^{3} T^{2} \)
73 \( 1 + 763 T + p^{3} T^{2} \)
79 \( 1 - 510 T + p^{3} T^{2} \)
83 \( 1 + 777 T + p^{3} T^{2} \)
89 \( 1 - 945 T + p^{3} T^{2} \)
97 \( 1 - 1246 T + p^{3} T^{2} \)
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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

−11.27856174220349058157076922998, −10.17213232277731410724179854550, −9.254912150699130830702310221281, −8.543292530951009284779640037331, −7.43336354897359897108373167150, −6.17641697460274263208818221288, −4.75611059288169044369471395799, −3.88057586572518254425850795176, −1.80151696573087936278175360327, 0, 1.80151696573087936278175360327, 3.88057586572518254425850795176, 4.75611059288169044369471395799, 6.17641697460274263208818221288, 7.43336354897359897108373167150, 8.543292530951009284779640037331, 9.254912150699130830702310221281, 10.17213232277731410724179854550, 11.27856174220349058157076922998

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