Properties

Label 2-15e2-1.1-c5-0-28
Degree $2$
Conductor $225$
Sign $-1$
Analytic cond. $36.0863$
Root an. cond. $6.00719$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 32·4-s + 25·7-s + 775·13-s + 1.02e3·16-s − 1.71e3·19-s − 800·28-s + 2.72e3·31-s − 1.65e4·37-s − 2.24e4·43-s − 1.61e4·49-s − 2.48e4·52-s + 5.69e4·61-s − 3.27e4·64-s − 7.34e4·67-s + 1.45e3·73-s + 5.47e4·76-s − 1.00e5·79-s + 1.93e4·91-s − 1.77e5·97-s − 1.40e5·103-s + 1.33e5·109-s + 2.56e4·112-s + ⋯
L(s)  = 1  − 4-s + 0.192·7-s + 1.27·13-s + 16-s − 1.08·19-s − 0.192·28-s + 0.508·31-s − 1.98·37-s − 1.85·43-s − 0.962·49-s − 1.27·52-s + 1.95·61-s − 64-s − 1.99·67-s + 0.0318·73-s + 1.08·76-s − 1.81·79-s + 0.245·91-s − 1.91·97-s − 1.30·103-s + 1.07·109-s + 0.192·112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(36.0863\)
Root analytic conductor: \(6.00719\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 225,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + p^{5} T^{2} \)
7 \( 1 - 25 T + p^{5} T^{2} \)
11 \( 1 + p^{5} T^{2} \)
13 \( 1 - 775 T + p^{5} T^{2} \)
17 \( 1 + p^{5} T^{2} \)
19 \( 1 + 1711 T + p^{5} T^{2} \)
23 \( 1 + p^{5} T^{2} \)
29 \( 1 + p^{5} T^{2} \)
31 \( 1 - 2723 T + p^{5} T^{2} \)
37 \( 1 + 16550 T + p^{5} T^{2} \)
41 \( 1 + p^{5} T^{2} \)
43 \( 1 + 22475 T + p^{5} T^{2} \)
47 \( 1 + p^{5} T^{2} \)
53 \( 1 + p^{5} T^{2} \)
59 \( 1 + p^{5} T^{2} \)
61 \( 1 - 56927 T + p^{5} T^{2} \)
67 \( 1 + 73475 T + p^{5} T^{2} \)
71 \( 1 + p^{5} T^{2} \)
73 \( 1 - 1450 T + p^{5} T^{2} \)
79 \( 1 + 100564 T + p^{5} T^{2} \)
83 \( 1 + p^{5} T^{2} \)
89 \( 1 + p^{5} T^{2} \)
97 \( 1 + 177725 T + p^{5} T^{2} \)
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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

−10.77995784373186041546310499609, −9.925979660163972277411813403660, −8.692624502105831220640237477844, −8.295656358778636899901034091369, −6.74304981344413844976704359070, −5.57462453937571278724762066698, −4.44723279043932538266830743420, −3.40797398115610612264803115084, −1.50201811755038771010386861744, 0, 1.50201811755038771010386861744, 3.40797398115610612264803115084, 4.44723279043932538266830743420, 5.57462453937571278724762066698, 6.74304981344413844976704359070, 8.295656358778636899901034091369, 8.692624502105831220640237477844, 9.925979660163972277411813403660, 10.77995784373186041546310499609

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