Properties

Label 2-15e2-1.1-c5-0-4
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 $0$

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: \(0\)
Selberg data: \((2,\ 225,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.101216651\)
\(L(\frac12)\) \(\approx\) \(1.101216651\)
\(L(\frac{7}{2})\) 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

−11.39825888653071464955936702706, −10.15556192059622741447063691491, −9.501144260336547159871356801912, −8.479406606271298693377812718209, −7.50176172819303334444570070550, −6.15947479967986932917829276875, −4.94064966072989435110134616222, −4.04120999774522986421201588952, −2.50144916854831244380728391478, −0.62152887550519764345145062023, 0.62152887550519764345145062023, 2.50144916854831244380728391478, 4.04120999774522986421201588952, 4.94064966072989435110134616222, 6.15947479967986932917829276875, 7.50176172819303334444570070550, 8.479406606271298693377812718209, 9.501144260336547159871356801912, 10.15556192059622741447063691491, 11.39825888653071464955936702706

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