Properties

Label 2-15e2-1.1-c5-0-18
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  − 4·2-s − 16·4-s − 225·7-s + 192·8-s + 434·11-s + 613·13-s + 900·14-s − 256·16-s + 878·17-s − 731·19-s − 1.73e3·22-s − 2.85e3·23-s − 2.45e3·26-s + 3.60e3·28-s + 7.58e3·29-s + 2.17e3·31-s − 5.12e3·32-s − 3.51e3·34-s − 9.31e3·37-s + 2.92e3·38-s + 1.20e4·41-s − 1.12e3·43-s − 6.94e3·44-s + 1.14e4·46-s − 2.98e4·47-s + 3.38e4·49-s − 9.80e3·52-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 1.73·7-s + 1.06·8-s + 1.08·11-s + 1.00·13-s + 1.22·14-s − 1/4·16-s + 0.736·17-s − 0.464·19-s − 0.764·22-s − 1.12·23-s − 0.711·26-s + 0.867·28-s + 1.67·29-s + 0.406·31-s − 0.883·32-s − 0.521·34-s − 1.11·37-s + 0.328·38-s + 1.11·41-s − 0.0924·43-s − 0.540·44-s + 0.794·46-s − 1.97·47-s + 2.01·49-s − 0.503·52-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 \( 1 + p^{2} T + p^{5} T^{2} \)
7 \( 1 + 225 T + p^{5} T^{2} \)
11 \( 1 - 434 T + p^{5} T^{2} \)
13 \( 1 - 613 T + p^{5} T^{2} \)
17 \( 1 - 878 T + p^{5} T^{2} \)
19 \( 1 + 731 T + p^{5} T^{2} \)
23 \( 1 + 2850 T + p^{5} T^{2} \)
29 \( 1 - 7582 T + p^{5} T^{2} \)
31 \( 1 - 2175 T + p^{5} T^{2} \)
37 \( 1 + 9310 T + p^{5} T^{2} \)
41 \( 1 - 12040 T + p^{5} T^{2} \)
43 \( 1 + 1121 T + p^{5} T^{2} \)
47 \( 1 + 29878 T + p^{5} T^{2} \)
53 \( 1 + 5740 T + p^{5} T^{2} \)
59 \( 1 - 5174 T + p^{5} T^{2} \)
61 \( 1 + 38717 T + p^{5} T^{2} \)
67 \( 1 - 31707 T + p^{5} T^{2} \)
71 \( 1 + 64472 T + p^{5} T^{2} \)
73 \( 1 + 19790 T + p^{5} T^{2} \)
79 \( 1 + 105000 T + p^{5} T^{2} \)
83 \( 1 - 3318 T + p^{5} T^{2} \)
89 \( 1 - 65376 T + p^{5} T^{2} \)
97 \( 1 + 919 p T + p^{5} T^{2} \)
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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

−10.46395057964099782189404256209, −9.832622123636126473033308178174, −9.030204884311004616049967836425, −8.181553270681953792484471355943, −6.76935879817766080495594820718, −6.00467863783875147592638660108, −4.24993366113546085051748188422, −3.27629961937534703941079798116, −1.26058785239748691981688964085, 0, 1.26058785239748691981688964085, 3.27629961937534703941079798116, 4.24993366113546085051748188422, 6.00467863783875147592638660108, 6.76935879817766080495594820718, 8.181553270681953792484471355943, 9.030204884311004616049967836425, 9.832622123636126473033308178174, 10.46395057964099782189404256209

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